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Small Ramsey numbers R(k, l)

Our own arrangement of the two-colour classical Ramsey numbers R(k, l) with 3 ≤ k ≤ 10 and k ≤ l ≤ 15, which is 72 of those 76 pairs because the survey Small Ramsey Numbers gives no value for R(9, 14), R(9, 15), R(10, 13) or R(10, 14). Each value is the one the survey prints, shown with the reference the survey credits for it, and marked either independently sourced, where our own check of what that reference credits supports the value, or credited to the survey and not independently checked. Every value marked the second way says why beside it, and the notes kept on each reading say what was actually read or recomputed.

The source. Stanisław Radziszowski, Small Ramsey Numbers, The Electronic Journal of Combinatorics, dynamic survey DS1, revision #18, 24 April 2026, Section 2.1, Tables Ia and Ib. The survey is credited for every value here and remains the authority. This page is our arrangement of its values and reference keys, not a copy of the document. What this page copies from the survey, and how to ask for a change.

What is covered. Section 2.1 only: 72 values of R(k, l). Table Ia prints something for 72 of them, and Table Ib prints a newer upper bound for 50. Not covered: everything else in the survey, including other graphs, more than two colours and hypergraphs, and any result published after revision #18.

What is watched. Once a day a scheduled job checks the journal’s page and the author’s revision list for a newer revision, and re-reads this table from the survey’s PDF, whose fingerprint is pinned. A new revision turns that job red until a person has compared it with this page. The run history is the live read; this page is not.

What has been checked against its source. The survey credits each bound to a paper by the key shown beside it, such as [AnM4], and an entry can credit its lower and upper bound to different papers. So every bound carries its own mark. Every bound Table Ia credits to a reference was taken in a census, completed on 15 October 2026: the 122 bounds credited under 42 reference keys were each attempted, and the note kept on each records what was tried. What a key credits is usually a paper, but can be a thesis, a manuscript, a personal communication, a web page, a published graph file, or a derivation the survey gives itself. A bound is marked independently sourced only when the reading was SOUND: what the key credits, or a construction or derivation it gives, was reached and supports the value; the notes kept on each reading say which edition or file that was. Every other bound is marked credited to the survey and not independently checked, with a sentence saying why; that includes bounds whose value was seen only in an abstract or in a different edition from the one the survey cites. A bound whose credited source was read and does not support it would carry neither mark; there are none. 6 values Table Ia prints credit no source to read (no reference is printed, or the reference is an item of the survey’s own Section 2.3), and Table Ib’s upper bounds were outside the census; the one Table Ib value the survey credits to a single paper is marked by that paper’s reading of the same bound in Table Ia. The notes kept on each reading name what was read and what it said. So far, of the 128 bounds Table Ia prints (its 124 numbers, with an exact value counted twice where its lower and upper halves are credited to different references): sound 53 (the credited source was read, and it supports this bound) · defective 0 (the credited source was read, and it does not support this bound) · unresolved 49 (what could be read did not settle this bound) · unreachable 20 (the credited source could not be read at all) · not yet read 0 · nothing to read it against 6. In all, 53 are independently sourced and 75 are credited to the survey and not independently checked.

Table Ia and Table Ib can print different upper bounds for the same number. Table Ib lists upper bounds from computations by Angeltveit and McKay ([AnM2], [AnM3], [AnM4]) and R(4, 5) from [MR4], and the survey says they improve on the upper bounds in Table Ia. Where both tables print one, this page shows both and does not choose between them for you.

[HW+] is the survey’s own abbreviation of [HWSYZH], as enhanced in [Boza5]. A reference written as item 2.3.h of the survey points to a numbered item of its Section 2.3, not to a paper.

If an entry disagrees with the survey or with the paper it credits, use looks wrong? on that entry. The report arrives already naming the entry and the revision.

k = 3

R(3, 3)

Table Ia prints
R(3, 3) = 6no reference printed
Table Ib prints
no entry in Table Ib
Read against its credited source
the exact value 6, with no reference printed: credited to the survey and not independently checked — nothing to read it against
looks wrong?

R(3, 4)

Table Ia prints
R(3, 4) = 9credited to [GG]
Table Ib prints
no entry in Table Ib
Read against its credited source
the exact value 9, credited to [GG]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
looks wrong?

R(3, 5)

Table Ia prints
R(3, 5) = 14credited to [GG]
Table Ib prints
no entry in Table Ib
Read against its credited source
the exact value 14, credited to [GG]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
looks wrong?

R(3, 6)

Table Ia prints
R(3, 6) = 18credited to [Kéry]
Table Ib prints
no entry in Table Ib
Read against its credited source
the exact value 18, credited to [Kéry]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
looks wrong?

R(3, 7)

Table Ia prints
R(3, 7) = 23lower bound credited to [Ka2] · upper bound credited to [GrY]
Table Ib prints
no entry in Table Ib
Read against its credited source
the lower bound 23, credited to [Ka2]: credited to the survey and not independently checked — unreachable: the credited source could not be read at all. The credited 1966 thesis could not be read: catalogue and library records of it were found, but none offered us its text (no preview, search-only access, a login, or a verification challenge that was not completed). How it was read
the upper bound 23, credited to [GrY]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. Only the abstract could be read, and it prints no number; the publisher blocked every attempt to open the paper itself. How it was read
looks wrong?

R(3, 8)

Table Ia prints
R(3, 8) = 28lower bound credited to [GR] · upper bound credited to [McZ]
Table Ib prints
no entry in Table Ib
Read against its credited source
the lower bound 28, credited to [GR]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
the upper bound 28, credited to [McZ]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
looks wrong?

R(3, 9)

Table Ia prints
R(3, 9) = 36lower bound credited to [Ka2] · upper bound credited to [GR]
Table Ib prints
no entry in Table Ib
Read against its credited source
the lower bound 36, credited to [Ka2]: credited to the survey and not independently checked — unreachable: the credited source could not be read at all. The credited 1966 thesis could not be read: catalogue and library records of it were found, but none offered us its text (no preview, search-only access, a login, or a verification challenge that was not completed). How it was read
the upper bound 36, credited to [GR]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
looks wrong?

R(3, 10)

Table Ia prints
40 ≤ R(3, 10) ≤ 41lower bound credited to [Ex5] · upper bound credited to [Ang1]
Table Ib prints
no entry in Table Ib
Read against its credited source
the lower bound 40, credited to [Ex5]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. Only the abstract could be read, from an archived copy of the publisher's page; it says a new lower bound for R(3, 10) is given but prints no value. How it was read
the upper bound 41, credited to [Ang1]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
looks wrong?

R(3, 11)

Table Ia prints
47 ≤ R(3, 11) ≤ 50lower bound credited to [Ex20] · upper bound credited to [GoeR1]
Table Ib prints
no entry in Table Ib
Read against its credited source
the lower bound 47, credited to [Ex20]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
the upper bound 50, credited to [GoeR1]: independently sourced — sound: the credited source was read, and it supports this bound. The paper is open access and was read: it states this bound in its abstract and again in Theorem 7. How it was read
looks wrong?

R(3, 12)

Table Ia prints
53 ≤ R(3, 12) ≤ 59lower bound credited to [Kol1] · upper bound credited to [Les]
Table Ib prints
no entry in Table Ib
Read against its credited source
the lower bound 53, credited to [Kol1]: credited to the survey and not independently checked — unreachable: the credited source could not be read at all. No text of the paper, not even an abstract, could be found: the publisher's online library begins with a later issue. How it was read
the upper bound 59, credited to [Les]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
looks wrong?

R(3, 13)

Table Ia prints
61 ≤ R(3, 13) ≤ 68lower bound credited to [NaRT] · upper bound credited to [GoeR1]
Table Ib prints
no entry in Table Ib
Read against its credited source
the lower bound 61, credited to [NaRT]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
the upper bound 68, credited to [GoeR1]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
looks wrong?

R(3, 14)

Table Ia prints
67 ≤ R(3, 14) ≤ 77lower bound credited to [Kol2] · upper bound credited to [GoeR1]
Table Ib prints
no entry in Table Ib
Read against its credited source
the lower bound 67, credited to [Kol2]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. The web page the survey cites states this bound in archived copies from 2022 to 2025, but no copy from 2016, the year the survey gives, was found. How it was read
the upper bound 77, credited to [GoeR1]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
looks wrong?

R(3, 15)

Table Ia prints
74 ≤ R(3, 15) ≤ 87lower bound credited to [Kol2] · upper bound credited to [GoeR1]
Table Ib prints
no entry in Table Ib
Read against its credited source
the lower bound 74, credited to [Kol2]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. The web page the survey cites states this bound in archived copies from 2022 to 2025, but no copy from 2016, the year the survey gives, was found. How it was read
the upper bound 87, credited to [GoeR1]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
looks wrong?

k = 4

R(4, 4)

Table Ia prints
R(4, 4) = 18credited to [GG]
Table Ib prints
no entry in Table Ib
Read against its credited source
the exact value 18, credited to [GG]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
looks wrong?

R(4, 5)

Table Ia prints
R(4, 5) = 25lower bound credited to [Ka1] · upper bound credited to [MR4]
Table Ib prints
R(4, 5) ≤ 25independently sourced: the survey credits this value to [MR4], and the same bound, credited to [MR4] in Table Ia, was found there (see the reading below)
Read against its credited source
the lower bound 25, credited to [Ka1]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
the upper bound 25, credited to [MR4]: independently sourced — sound: the credited source was read, and it supports this bound. Read in the authors' own copy of the paper, not the journal's typeset edition. How it was read
looks wrong?

R(4, 6)

Table Ia prints
36 ≤ R(4, 6) ≤ 41lower bound credited to [Ex19] · upper bound credited to [MR5]
Table Ib prints
R(4, 6) ≤ 40credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 36, credited to [Ex19]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
the upper bound 41, credited to [MR5]: independently sourced — sound: the credited source was read, and it supports this bound. The journal's abstract states this bound, and its proof was read in the authors' preprint rather than the journal edition. How it was read
looks wrong?

R(4, 7)

Table Ia prints
49 ≤ R(4, 7) ≤ 61lower bound credited to [Ex3] · upper bound credited to [Mac]
Table Ib prints
R(4, 7) ≤ 58credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 49, credited to [Ex3]: credited to the survey and not independently checked — unreachable: the credited source could not be read at all. No text of the cited book chapter could be reached: the copies found are lent only to account holders or can only be searched. How it was read
the upper bound 61, credited to [Mac]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. The credited thesis prints this value in a table without deriving it, and says the table was compiled by the survey's own author, so it was not counted as an independent source for this value. How it was read
looks wrong?

R(4, 8)

Table Ia prints
59 ≤ R(4, 8) ≤ 84lower bound credited to [ExT] · upper bound credited to [Mac]
Table Ib prints
R(4, 8) ≤ 79credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 59, credited to [ExT]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
the upper bound 84, credited to [Mac]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. The credited thesis prints this value in a table without deriving it, and says the table was compiled by the survey's own author, so it was not counted as an independent source for this value. How it was read
looks wrong?

R(4, 9)

Table Ia prints
73 ≤ R(4, 9) ≤ 115lower bound credited to [Ex16] · upper bound credited to [Mac]
Table Ib prints
R(4, 9) ≤ 105credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 73, credited to [Ex16]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. The survey credits a personal communication, which has no public form. Archived copies of the web page it points to list this bound, and the colouring the page links today passed our computer check; neither shows what the communication itself contained. How it was read
the upper bound 115, credited to [Mac]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. The credited thesis prints this value in a table without deriving it, and says the table was compiled by the survey's own author, so it was not counted as an independent source for this value. How it was read
looks wrong?

R(4, 10)

Table Ia prints
92 ≤ R(4, 10) ≤ 149lower bound credited to [HaKr1] · upper bound credited to [Mac]
Table Ib prints
R(4, 10) ≤ 135credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 92, credited to [HaKr1]: independently sourced — sound: the credited source was read, and it supports this bound. The journal volume was read only as short searchable text extracts, which state this bound; no image of a page carrying it was seen. How it was read
the upper bound 149, credited to [Mac]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. The credited thesis prints this value in a table without deriving it, and says the table was compiled by the survey's own author, so it was not counted as an independent source for this value. How it was read
looks wrong?

R(4, 11)

Table Ia prints
102 ≤ R(4, 11) ≤ 191lower bound credited to [ExT] · upper bound credited to [Spe4]
Table Ib prints
R(4, 11) ≤ 170credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 102, credited to [ExT]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
the upper bound 191, credited to [Spe4]: credited to the survey and not independently checked — unreachable: the credited source could not be read at all. The survey credits a 1993 personal communication and a 1994 manuscript; no public copy of either was found. How it was read
looks wrong?

R(4, 12)

Table Ia prints
128 ≤ R(4, 12) ≤ 238lower bound credited to [SuLL] · upper bound credited to [Spe4]
Table Ib prints
R(4, 12) ≤ 210credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 128, credited to [SuLL]: credited to the survey and not independently checked — unreachable: the credited source could not be read at all. The publisher shows the paper as subscription content with no abstract, and the searches recorded found no open copy, so nothing that states the value was read. How it was read
the upper bound 238, credited to [Spe4]: credited to the survey and not independently checked — unreachable: the credited source could not be read at all. The survey credits a 1993 personal communication and a 1994 manuscript; no public copy of either was found. How it was read
looks wrong?

R(4, 13)

Table Ia prints
139 ≤ R(4, 13) ≤ 291lower bound credited to [NaRT] · upper bound credited to [Spe4]
Table Ib prints
R(4, 13) ≤ 256credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 139, credited to [NaRT]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
the upper bound 291, credited to [Spe4]: credited to the survey and not independently checked — unreachable: the credited source could not be read at all. The survey credits a 1993 personal communication and a 1994 manuscript; no public copy of either was found. How it was read
looks wrong?

R(4, 14)

Table Ia prints
148 ≤ R(4, 14) ≤ 349lower bound credited to [NaRT] · upper bound credited to [Spe4]
Table Ib prints
R(4, 14) ≤ 307credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 148, credited to [NaRT]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
the upper bound 349, credited to [Spe4]: credited to the survey and not independently checked — unreachable: the credited source could not be read at all. The survey credits a 1993 personal communication and a 1994 manuscript; no public copy of either was found. How it was read
looks wrong?

R(4, 15)

Table Ia prints
159 ≤ R(4, 15) ≤ 417lower bound credited to [NaRT] · upper bound credited to [Spe4]
Table Ib prints
R(4, 15) ≤ 364credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 159, credited to [NaRT]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
the upper bound 417, credited to [Spe4]: credited to the survey and not independently checked — unreachable: the credited source could not be read at all. The survey credits a 1993 personal communication and a 1994 manuscript; no public copy of either was found. How it was read
looks wrong?

k = 5

R(5, 5)

Table Ia prints
43 ≤ R(5, 5) ≤ 46lower bound credited to [Ex4] · upper bound credited to [AnM4]
Table Ib prints
R(5, 5) ≤ 46credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 43, credited to [Ex4]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. Only the abstract could be read; it states this bound, but the body, where the construction is given, could not be opened. The colouring the author links passed our computer check, which is not a reading of the paper. How it was read
the upper bound 46, credited to [AnM4]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. The authors' arXiv preprint states this bound, but the journal edition the survey cites could not be opened, so its theorem could not be compared. How it was read
looks wrong?

R(5, 6)

Table Ia prints
59 ≤ R(5, 6) ≤ 87lower bound credited to [Ex25] · upper bound credited to [HZ1]
Table Ib prints
R(5, 6) ≤ 85credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 59, credited to [Ex25]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. An arXiv preprint states this bound, but the survey cites a 2023 manuscript, and the preprint was not shown to be that manuscript. How it was read
the upper bound 87, credited to [HZ1]: credited to the survey and not independently checked — unreachable: the credited source could not be read at all. No copy of the credited 1993 paper, and no catalogue entry for it, was found. How it was read
looks wrong?

R(5, 7)

Table Ia prints
80 ≤ R(5, 7) ≤ 143lower bound credited to [CaET] · upper bound credited to [HZ1]
Table Ib prints
R(5, 7) ≤ 133credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 80, credited to [CaET]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. The authors' manuscript states this bound, but the journal edition the survey cites could not be opened. How it was read
the upper bound 143, credited to [HZ1]: credited to the survey and not independently checked — unreachable: the credited source could not be read at all. No copy of the credited 1993 paper, and no catalogue entry for it, was found. How it was read
looks wrong?

R(5, 8)

Table Ia prints
101 ≤ R(5, 8) ≤ 216lower bound credited to [HaKr1] · upper bound credited to [Spe4]
Table Ib prints
R(5, 8) ≤ 193credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 101, credited to [HaKr1]: independently sourced — sound: the credited source was read, and it supports this bound. The journal volume was read only as short searchable text extracts, which state this bound; no image of a page carrying it was seen. How it was read
the upper bound 216, credited to [Spe4]: credited to the survey and not independently checked — unreachable: the credited source could not be read at all. The survey credits a 1993 personal communication and a 1994 manuscript; no public copy of either was found. How it was read
looks wrong?

R(5, 9)

Table Ia prints
133 ≤ R(5, 9) ≤ 316lower bound credited to [Kuz] · upper bound credited to [Mac]
Table Ib prints
R(5, 9) ≤ 282credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 133, credited to [Kuz]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
the upper bound 316, credited to [Mac]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. The credited thesis prints this value in a table without deriving it, and says the table was compiled by the survey's own author, so it was not counted as an independent source for this value. How it was read
looks wrong?

R(5, 10)

Table Ia prints
149 ≤ R(5, 10) ≤ 442lower bound credited to [ExT] · upper bound credited to [Mac]
Table Ib prints
R(5, 10) ≤ 381credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 149, credited to [ExT]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
the upper bound 442, credited to [Mac]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. The credited thesis prints this value in a table without deriving it, and says the table was compiled by the survey's own author, so it was not counted as an independent source for this value. How it was read
looks wrong?

R(5, 11)

Table Ia prints
183 ≤ R(5, 11) ≤ 633lower bound credited to [Kuz] · upper bound credited to [HW+]
Table Ib prints
R(5, 11) ≤ 511credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 183, credited to [Kuz]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
the upper bound 633, credited to [HW+]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. Only the paper's abstract could be read, and it does not name this value; the publisher blocked every attempt to open the paper itself. The survey's note 2.1.o says Boza computed bounds marked HW+ in 2013 and cites a personal communication for them; whether this value is one of those is not established here. How it was read
looks wrong?

R(5, 12)

Table Ia prints
203 ≤ R(5, 12) ≤ 848lower bound credited to [Kuz] · upper bound credited to [HW+]
Table Ib prints
R(5, 12) ≤ 672credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 203, credited to [Kuz]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
the upper bound 848, credited to [HW+]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. Only the paper's abstract could be read. It states this value, but the body, where the bound is derived, could not be opened: the publisher blocked every attempt. The survey's note 2.1.o says Boza computed bounds marked HW+ in 2013 and cites a personal communication for them; whether this value is one of those is not established here. How it was read
looks wrong?

R(5, 13)

Table Ia prints
233 ≤ R(5, 13) ≤ 1138lower bound credited to [Kuz] · upper bound credited to [HW+]
Table Ib prints
R(5, 13) ≤ 860credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 233, credited to [Kuz]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
the upper bound 1138, credited to [HW+]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. Only the paper's abstract could be read, and it does not name this value; the publisher blocked every attempt to open the paper itself. The survey's note 2.1.o says Boza computed bounds marked HW+ in 2013 and cites a personal communication for them; whether this value is one of those is not established here. How it was read
looks wrong?

R(5, 14)

Table Ia prints
267 ≤ R(5, 14) ≤ 1461lower bound credited to [Kuz] · upper bound credited to [HW+]
Table Ib prints
R(5, 14) ≤ 1081credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 267, credited to [Kuz]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
the upper bound 1461, credited to [HW+]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. Only the paper's abstract could be read. It states this value, but the body, where the bound is derived, could not be opened: the publisher blocked every attempt. The survey's note 2.1.o says Boza computed bounds marked HW+ in 2013 and cites a personal communication for them; whether this value is one of those is not established here. How it was read
looks wrong?

R(5, 15)

Table Ia prints
275 ≤ R(5, 15) ≤ 1878lower bound credited to item 2.3.h of the survey · upper bound credited to [HW+]
Table Ib prints
R(5, 15) ≤ 1341credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 275, credited to item 2.3.h of the survey: credited to the survey and not independently checked — nothing to read it against
the upper bound 1878, credited to [HW+]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. Only the paper's abstract could be read, and it does not name this value; the publisher blocked every attempt to open the paper itself. The survey's note 2.1.o says Boza computed bounds marked HW+ in 2013 and cites a personal communication for them; whether this value is one of those is not established here. How it was read
looks wrong?

k = 6

R(6, 6)

Table Ia prints
102 ≤ R(6, 6) ≤ 165lower bound credited to [Ka2] · upper bound credited to [Mac]
Table Ib prints
R(6, 6) ≤ 160credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 102, credited to [Ka2]: credited to the survey and not independently checked — unreachable: the credited source could not be read at all. The credited 1966 thesis could not be read: catalogue and library records of it were found, but none offered us its text (no preview, search-only access, a login, or a verification challenge that was not completed). How it was read
the upper bound 165, credited to [Mac]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
looks wrong?

R(6, 7)

Table Ia prints
115 ≤ R(6, 7) ≤ 298lower bound credited to [ExT] · upper bound credited to [HZ1]
Table Ib prints
R(6, 7) ≤ 270credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 115, credited to [ExT]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
the upper bound 298, credited to [HZ1]: credited to the survey and not independently checked — unreachable: the credited source could not be read at all. No copy of the credited 1993 paper, and no catalogue entry for it, was found. How it was read
looks wrong?

R(6, 8)

Table Ia prints
134 ≤ R(6, 8) ≤ 495lower bound credited to [ExT] · upper bound credited to [Mac]
Table Ib prints
R(6, 8) ≤ 423credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 134, credited to [ExT]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
the upper bound 495, credited to [Mac]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. The credited thesis prints this value in a table without deriving it, and says the table was compiled by the survey's own author, so it was not counted as an independent source for this value. How it was read
looks wrong?

R(6, 9)

Table Ia prints
183 ≤ R(6, 9) ≤ 780lower bound credited to [Kuz] · upper bound credited to [Mac]
Table Ib prints
R(6, 9) ≤ 651credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 183, credited to [Kuz]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
the upper bound 780, credited to [Mac]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. The credited thesis prints this value in a table without deriving it, and says the table was compiled by the survey's own author, so it was not counted as an independent source for this value. How it was read
looks wrong?

R(6, 10)

Table Ia prints
204 ≤ R(6, 10) ≤ 1171lower bound credited to [Kuz] · upper bound credited to [Mac]
Table Ib prints
R(6, 10) ≤ 944credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 204, credited to [Kuz]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
the upper bound 1171, credited to [Mac]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. The credited thesis prints this value in a table without deriving it, and says the table was compiled by the survey's own author, so it was not counted as an independent source for this value. How it was read
looks wrong?

R(6, 11)

Table Ia prints
262 ≤ R(6, 11) ≤ 1804lower bound credited to [Tat] · upper bound credited to [HW+]
Table Ib prints
R(6, 11) ≤ 1346credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 262, credited to [Tat]: independently sourced — sound: the credited source was read, and it supports this bound. The credited construction is a published graph file rather than a paper: it was downloaded and checked exhaustively, and it does prove this bound. How it was read
the upper bound 1804, credited to [HW+]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. Only the paper's abstract could be read, and it does not name this value; the publisher blocked every attempt to open the paper itself. The survey's note 2.1.o says Boza computed bounds marked HW+ in 2013 and cites a personal communication for them; whether this value is one of those is not established here. How it was read
looks wrong?

R(6, 12)

Table Ia prints
294 ≤ R(6, 12) ≤ 2566lower bound credited to [Kuz] · upper bound credited to [HW+]
Table Ib prints
R(6, 12) ≤ 1855credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 294, credited to [Kuz]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
the upper bound 2566, credited to [HW+]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. Only the paper's abstract could be read, and it does not name this value; the publisher blocked every attempt to open the paper itself. The survey's note 2.1.o says Boza computed bounds marked HW+ in 2013 and cites a personal communication for them; whether this value is one of those is not established here. How it was read
looks wrong?

R(6, 13)

Table Ia prints
347 ≤ R(6, 13) ≤ 3703lower bound credited to [Kuz] · upper bound credited to [HW+]
Table Ib prints
R(6, 13) ≤ 2499credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 347, credited to [Kuz]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
the upper bound 3703, credited to [HW+]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. Only the paper's abstract could be read, and it does not name this value; the publisher blocked every attempt to open the paper itself. The survey's note 2.1.o says Boza computed bounds marked HW+ in 2013 and cites a personal communication for them; whether this value is one of those is not established here. How it was read
looks wrong?

R(6, 14)

Table Ia prints
R(6, 14) ≤ 5033 (no lower bound printed)upper bound credited to [HW+]
Table Ib prints
R(6, 14) ≤ 3301credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the upper bound 5033, credited to [HW+]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. Only the paper's abstract could be read, and it does not name this value; the publisher blocked every attempt to open the paper itself. The survey's note 2.1.o says Boza computed bounds marked HW+ in 2013 and cites a personal communication for them; whether this value is one of those is not established here. How it was read
looks wrong?

R(6, 15)

Table Ia prints
401 ≤ R(6, 15) ≤ 6911lower bound credited to item 2.3.i of the survey · upper bound credited to [HW+]
Table Ib prints
R(6, 15) ≤ 4305credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 401, credited to item 2.3.i of the survey: credited to the survey and not independently checked — nothing to read it against
the upper bound 6911, credited to [HW+]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. Only the paper's abstract could be read, and it does not name this value; the publisher blocked every attempt to open the paper itself. The survey's note 2.1.o says Boza computed bounds marked HW+ in 2013 and cites a personal communication for them; whether this value is one of those is not established here. How it was read
looks wrong?

k = 7

R(7, 7)

Table Ia prints
205 ≤ R(7, 7) ≤ 540lower bound credited to [Math] · upper bound credited to [Mac]
Table Ib prints
R(7, 7) ≤ 492credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 205, credited to [Math]: credited to the survey and not independently checked — unreachable: the credited source could not be read at all. Nothing beyond the paper's catalogue entry could be reached; the publisher blocked every attempt to open it. How it was read
the upper bound 540, credited to [Mac]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
looks wrong?

R(7, 8)

Table Ia prints
219 ≤ R(7, 8) ≤ 1031lower bound credited to [Tat] · upper bound credited to [HZ1]
Table Ib prints
R(7, 8) ≤ 832credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 219, credited to [Tat]: independently sourced — sound: the credited source was read, and it supports this bound. The credited construction is a published graph file rather than a paper: it was downloaded and checked exhaustively, and it does prove this bound. How it was read
the upper bound 1031, credited to [HZ1]: credited to the survey and not independently checked — unreachable: the credited source could not be read at all. No copy of the credited 1993 paper, and no catalogue entry for it, was found. How it was read
looks wrong?

R(7, 9)

Table Ia prints
252 ≤ R(7, 9) ≤ 1713lower bound credited to [Kuz] · upper bound credited to [HZ2]
Table Ib prints
R(7, 9) ≤ 1368credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 252, credited to [Kuz]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
the upper bound 1713, credited to [HZ2]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
looks wrong?

R(7, 10)

Table Ia prints
292 ≤ R(7, 10) ≤ 2826lower bound credited to [Kuz] · upper bound credited to [Mac]
Table Ib prints
R(7, 10) ≤ 2119credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 292, credited to [Kuz]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
the upper bound 2826, credited to [Mac]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. The credited thesis prints this value in a table without deriving it, and says the table was compiled by the survey's own author, so it was not counted as an independent source for this value. How it was read
looks wrong?

R(7, 11)

Table Ia prints
405 ≤ R(7, 11) ≤ 4553lower bound credited to [XXER] · upper bound credited to [HW+]
Table Ib prints
R(7, 11) ≤ 3197credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 405, credited to [XXER]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
the upper bound 4553, credited to [HW+]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. Only the paper's abstract could be read, and it does not name this value; the publisher blocked every attempt to open the paper itself. The survey's note 2.1.o says Boza computed bounds marked HW+ in 2013 and cites a personal communication for them; whether this value is one of those is not established here. How it was read
looks wrong?

R(7, 12)

Table Ia prints
417 ≤ R(7, 12) ≤ 6954lower bound credited to [XSR2] · upper bound credited to [HW+]
Table Ib prints
R(7, 12) ≤ 4665credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 417, credited to [XSR2]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. The authors' preprint states this bound and the journal's abstract names it, but the journal edition the survey cites could not be opened. How it was read
the upper bound 6954, credited to [HW+]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. Only the paper's abstract could be read, and it does not name this value; the publisher blocked every attempt to open the paper itself. The survey's note 2.1.o says Boza computed bounds marked HW+ in 2013 and cites a personal communication for them; whether this value is one of those is not established here. How it was read
looks wrong?

R(7, 13)

Table Ia prints
511 ≤ R(7, 13) ≤ 10578lower bound credited to [XuXR] · upper bound credited to [HW+]
Table Ib prints
R(7, 13) ≤ 6653credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 511, credited to [XuXR]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. The authors' preprint states this bound, but the journal edition the survey cites could not be opened. How it was read
the upper bound 10578, credited to [HW+]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. Only the paper's abstract could be read, and it does not name this value; the publisher blocked every attempt to open the paper itself. The survey's note 2.1.o says Boza computed bounds marked HW+ in 2013 and cites a personal communication for them; whether this value is one of those is not established here. How it was read
looks wrong?

R(7, 14)

Table Ia prints
R(7, 14) ≤ 15263 (no lower bound printed)upper bound credited to [HW+]
Table Ib prints
R(7, 14) ≤ 9260credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the upper bound 15263, credited to [HW+]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. Only the paper's abstract could be read, and it does not name this value; the publisher blocked every attempt to open the paper itself. The survey's note 2.1.o says Boza computed bounds marked HW+ in 2013 and cites a personal communication for them; whether this value is one of those is not established here. How it was read
looks wrong?

R(7, 15)

Table Ia prints
R(7, 15) ≤ 22112 (no lower bound printed)upper bound credited to [HW+]
Table Ib prints
R(7, 15) ≤ 12635credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the upper bound 22112, credited to [HW+]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. Only the paper's abstract could be read, and it does not name this value; the publisher blocked every attempt to open the paper itself. The survey's note 2.1.o says Boza computed bounds marked HW+ in 2013 and cites a personal communication for them; whether this value is one of those is not established here. How it was read
looks wrong?

k = 8

R(8, 8)

Table Ia prints
282 ≤ R(8, 8) ≤ 1870lower bound credited to [BurR] · upper bound credited to [Mac]
Table Ib prints
R(8, 8) ≤ 1518credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 282, credited to [BurR]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. Only the abstract could be read, and it names no value; the publisher blocked every attempt to open the paper itself. How it was read
the upper bound 1870, credited to [Mac]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
looks wrong?

R(8, 9)

Table Ia prints
329 ≤ R(8, 9) ≤ 3583lower bound credited to [Kuz] · upper bound credited to [Ea1]
Table Ib prints
R(8, 9) ≤ 2662credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 329, credited to [Kuz]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
the upper bound 3583, credited to [Ea1]: independently sourced — sound: the credited source was read, and it supports this bound. The survey derives this value from two others in this table, R(7, 9) ≤ 1713 and R(8, 8) ≤ 1870, and both were found in the papers credited for them. How it was read
looks wrong?

R(8, 10)

Table Ia prints
343 ≤ R(8, 10) ≤ 6090lower bound credited to [Kuz] · upper bound credited to [HZ2]
Table Ib prints
R(8, 10) ≤ 4402credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 343, credited to [Kuz]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
the upper bound 6090, credited to [HZ2]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
looks wrong?

R(8, 11)

Table Ia prints
457 ≤ R(8, 11) ≤ 10630lower bound credited to item 2.3.i of the survey · upper bound credited to [HW+]
Table Ib prints
R(8, 11) ≤ 7040credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 457, credited to item 2.3.i of the survey: credited to the survey and not independently checked — nothing to read it against
the upper bound 10630, credited to [HW+]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. Only the paper's abstract could be read, and it does not name this value; the publisher blocked every attempt to open the paper itself. The survey's note 2.1.o says Boza computed bounds marked HW+ in 2013 and cites a personal communication for them; whether this value is one of those is not established here. How it was read
looks wrong?

R(8, 12)

Table Ia prints
R(8, 12) ≤ 16944 (no lower bound printed)upper bound credited to [HW+]
Table Ib prints
R(8, 12) ≤ 10836credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the upper bound 16944, credited to [HW+]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. Only the paper's abstract could be read, and it does not name this value; the publisher blocked every attempt to open the paper itself. The survey's note 2.1.o says Boza computed bounds marked HW+ in 2013 and cites a personal communication for them; whether this value is one of those is not established here. How it was read
looks wrong?

R(8, 13)

Table Ia prints
817 ≤ R(8, 13) ≤ 27485lower bound credited to [XXER] · upper bound credited to [HW+]
Table Ib prints
no entry in Table Ib
Read against its credited source
the lower bound 817, credited to [XXER]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
the upper bound 27485, credited to [HW+]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. Only the paper's abstract could be read, and it does not name this value; the publisher blocked every attempt to open the paper itself. The survey's note 2.1.o says Boza computed bounds marked HW+ in 2013 and cites a personal communication for them; whether this value is one of those is not established here. How it was read
looks wrong?

R(8, 14)

Table Ia prints
R(8, 14) ≤ 41525 (no lower bound printed)upper bound credited to [HW+]
Table Ib prints
no entry in Table Ib
Read against its credited source
the upper bound 41525, credited to [HW+]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. Only the paper's abstract could be read, and it does not name this value; the publisher blocked every attempt to open the paper itself. The survey's note 2.1.o says Boza computed bounds marked HW+ in 2013 and cites a personal communication for them; whether this value is one of those is not established here. How it was read
looks wrong?

R(8, 15)

Table Ia prints
873 ≤ R(8, 15) ≤ 63609lower bound credited to item 2.3.i of the survey · upper bound credited to [HW+]
Table Ib prints
no entry in Table Ib
Read against its credited source
the lower bound 873, credited to item 2.3.i of the survey: credited to the survey and not independently checked — nothing to read it against
the upper bound 63609, credited to [HW+]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. Only the paper's abstract could be read, and it does not name this value; the publisher blocked every attempt to open the paper itself. The survey's note 2.1.o says Boza computed bounds marked HW+ in 2013 and cites a personal communication for them; whether this value is one of those is not established here. How it was read
looks wrong?

k = 9

R(9, 9)

Table Ia prints
565 ≤ R(9, 9) ≤ 6588lower bound credited to [Math] · upper bound credited to [ShZ1]
Table Ib prints
R(9, 9) ≤ 4956credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 565, credited to [Math]: credited to the survey and not independently checked — unreachable: the credited source could not be read at all. Nothing beyond the paper's catalogue entry could be reached; the publisher blocked every attempt to open it. How it was read
the upper bound 6588, credited to [ShZ1]: credited to the survey and not independently checked — unreachable: the credited source could not be read at all. No copy of the credited 2001 manuscript was found; the only trace is a bibliography line in another source that cites it. How it was read
looks wrong?

R(9, 10)

Table Ia prints
581 ≤ R(9, 10) ≤ 12677lower bound credited to [XSR2] · upper bound credited to [Ea1]
Table Ib prints
R(9, 10) ≤ 8675credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 581, credited to [XSR2]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. The authors' preprint states this bound and the journal's abstract names it, but the journal edition the survey cites could not be opened. How it was read
the upper bound 12677, credited to [Ea1]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. The survey derives this value from two others in this table, and one of them, R(9, 9) ≤ 6588, could not be checked against the paper credited for it. How it was read
looks wrong?

R(9, 11)

Table Ia prints
R(9, 11) ≤ 22325 (no lower bound printed)upper bound credited to [HW+]
Table Ib prints
R(9, 11) ≤ 14631credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the upper bound 22325, credited to [HW+]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. Only the paper's abstract could be read, and it does not name this value; the publisher blocked every attempt to open the paper itself. The survey's note 2.1.o says Boza computed bounds marked HW+ in 2013 and cites a personal communication for them; whether this value is one of those is not established here. How it was read
looks wrong?

R(9, 12)

Table Ia prints
R(9, 12) ≤ 38832 (no lower bound printed)upper bound credited to [HW+]
Table Ib prints
no entry in Table Ib
Read against its credited source
the upper bound 38832, credited to [HW+]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. Only the paper's abstract could be read, and it does not name this value; the publisher blocked every attempt to open the paper itself. The survey's note 2.1.o says Boza computed bounds marked HW+ in 2013 and cites a personal communication for them; whether this value is one of those is not established here. How it was read
looks wrong?

R(9, 13)

Table Ia prints
R(9, 13) ≤ 64864 (no lower bound printed)upper bound credited to [HW+]
Table Ib prints
no entry in Table Ib
Read against its credited source
the upper bound 64864, credited to [HW+]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. Only the paper's abstract could be read, and it does not name this value; the publisher blocked every attempt to open the paper itself. The survey's note 2.1.o says Boza computed bounds marked HW+ in 2013 and cites a personal communication for them; whether this value is one of those is not established here. How it was read
looks wrong?

k = 10

R(10, 10)

Table Ia prints
798 ≤ R(10, 10) ≤ 23556lower bound credited to [Math] · upper bound credited to [Shi2]
Table Ib prints
R(10, 10) ≤ 16064credited to the survey and not independently checked: the census read Table Ia only; the survey credits Table Ib's values as a group to [AnM2], [AnM3] and [AnM4], not each to one paper
Read against its credited source
the lower bound 798, credited to [Math]: credited to the survey and not independently checked — unreachable: the credited source could not be read at all. Nothing beyond the paper's catalogue entry could be reached; the publisher blocked every attempt to open it. How it was read
the upper bound 23556, credited to [Shi2]: independently sourced — sound: the credited source was read, and it supports this bound. How it was read
looks wrong?

R(10, 11)

Table Ia prints
R(10, 11) ≤ 45881 (no lower bound printed)upper bound credited to [HW+]
Table Ib prints
no entry in Table Ib
Read against its credited source
the upper bound 45881, credited to [HW+]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. Only the paper's abstract could be read, and it does not name this value; the publisher blocked every attempt to open the paper itself. The survey's note 2.1.o says Boza computed bounds marked HW+ in 2013 and cites a personal communication for them; whether this value is one of those is not established here. How it was read
looks wrong?

R(10, 12)

Table Ia prints
R(10, 12) ≤ 81123 (no lower bound printed)upper bound credited to [HW+]
Table Ib prints
no entry in Table Ib
Read against its credited source
the upper bound 81123, credited to [HW+]: credited to the survey and not independently checked — unresolved: what could be read did not settle this bound. Only the paper's abstract could be read, and it does not name this value; the publisher blocked every attempt to open the paper itself. The survey's note 2.1.o says Boza computed bounds marked HW+ in 2013 and cites a personal communication for them; whether this value is one of those is not established here. How it was read
looks wrong?

R(10, 15)

Table Ia prints
R(10, 15) ≥ 1313 (no upper bound printed)lower bound credited to item 2.3.i of the survey
Table Ib prints
no entry in Table Ib
Read against its credited source
the lower bound 1313, credited to item 2.3.i of the survey: credited to the survey and not independently checked — nothing to read it against
looks wrong?