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Small Ramsey numbers — table of known values and bounds (2026)

The known values and best bounds for the two-colour Ramsey numbers R(s,t) with s, t ≤ 10, as of the April 2026 revision of Radziszowski's survey — with who proved what, and which entries are open to new constructions.

Updated 2026-10-03 · CC BY 4.0

The Ramsey number R(s,t) is the smallest n such that every red/blue colouring of the edges of the complete graph K_n contains a red K_s or a blue K_t. Equivalently, R(s,t) − 1 is the largest number of vertices of a graph with no clique of size s and no independent set of size t. Ramsey's theorem says these numbers exist; computing them is notoriously hard. Only nine non-trivial two-colour values are known.

The standard reference is Stanisław Radziszowski's dynamic survey Small Ramsey Numbers in the Electronic Journal of Combinatorics; the table below follows its revision 18 of 24 April 2026 (current version). We re-check it when the survey is revised.

The table

A single number is an exact value; a range a–b means a ≤ R(s,t) ≤ b.

s \ t345678910
369141823283640–41
4182536–4049–5859–7973–10592–135
543–4659–8580–133101–193133–282149–381
6102–160115–270134–423183–651204–944
7205–492219–832252–1368292–2119
8282–1518329–2662343–4402
9565–4956581–8675
10798–16064

R(s,t) = R(t,s), so only the upper triangle is shown. R(2,t) = t trivially.

The known values

  • R(3,3) = 6, R(3,4) = 9, R(3,5) = 14, R(4,4) = 18 — Greenwood and Gleason (1955).
  • R(3,6) = 18 — Kéry (1964).
  • R(3,7) = 23 — Graver and Yackel (1968).
  • R(3,8) = 28 — McKay and Zhang Ke Min (1992).
  • R(3,9) = 36 — Grinstead and Roberts (1982).
  • R(4,5) = 25 — McKay and Radziszowski (1995), by a large computer search.

Every exact value beyond R(4,4) needed substantial computation. No new two-colour value has been determined since 1995.

Recent changes to the bounds

Upper bounds. Most of the upper bounds for s ≥ 4 in the table come from large computations by Vigleik Angeltveit and Brendan McKay between 2019 and 2024, which combine counting identities, gluing of smaller Ramsey graphs and linear programming. Two highlights:

  • R(5,5) ≤ 46 (Angeltveit–McKay, arXiv 2409.15709, published in the Journal of Graph Theory in 2026), down from 48. The lower bound 43 is Exoo's construction from 1989, so 43 ≤ R(5,5) ≤ 46. McKay and Radziszowski conjectured in 1997 that the true value is 43.
  • R(4,6) ≤ 40, down from 41 (McKay–Radziszowski). See the R(4,6) guide for the history.

R(3,10). Angeltveit proved R(3,10) ≤ 41 (arXiv 2401.00392, Electronic Journal of Combinatorics 2025), leaving only two possible values: 40 or 41.

Lower bounds come from explicit colourings, found by algebraic constructions (circulant and Paley-type graphs) or by local search. They change less often but are the easiest place for a newcomer to contribute: a colouring is a certificate that anyone can check in a fraction of a second.

How to check a claimed lower bound

A lower bound R(s,t) > n is proved by a graph on n vertices with no K_s and no independent set of size t. Checking it is a finite search: enumerate all s-subsets and t-subsets (or use a clique finder) and confirm none is a clique / independent set. For n around 40 this takes milliseconds to seconds.

On Cairn Commons, the R(4,6) and R(5,5) problems accept lower-bound certificates directly: an adjacency matrix, checked automatically by the Ramsey checker. A colouring of K_36 for R(4,6), or of K_43 for R(5,5), would be a new result.

How upper bounds are proved

Upper bounds need an argument that no colouring of K_n avoids both structures. For small cases this is done by:

  • Counting and degree arguments: in a (s,t)-colouring of K_n, each vertex's red and blue neighbourhoods are themselves Ramsey colourings for (s−1,t) and (s,t−1).
  • Gluing: enumerate all possible neighbourhoods (from catalogues of smaller Ramsey graphs) and show they cannot be glued together.
  • Linear and integer programming on counts of substructures.
  • SAT solving with proof certificates, increasingly used for related problems such as the Schur number S(5) = 160 (2017).

Upper-bound work is reviewed on Cairn Commons as a computation: code, intermediate data and certificates, re-run by a reviewer.

The same "how large before order appears" question gives the Schur numbers (sum-free colourings of integers) and the van der Waerden numbers (monochromatic arithmetic progressions). The Ramsey theory topic page lists all open Ramsey-type problems in the catalogue, including dozens of Erdős problems.

Sources

Problems mentioned in this guide

  • The Ramsey number R(4,6) — Narrow the gap 36 ≤ R(4,6) ≤ 40. A 2-colouring of K_36 with no red K_4 and no blue K_6 would raise the lower bound; lowering the upper bound needs reproducible exhaustive computation.
  • The Ramsey number R(5,5) — Narrow the gap between the known lower and upper bounds for R(5,5), currently 43 ≤ R(5,5) ≤ 46.
  • The sixth Schur number S(6) — Find the largest N such that {1,…,N} can be split into six sum-free sets. After Heule's 2017 SAT proof that S(5) = 160, the best known bound is S(6) ≥ 536, with a large gap to the upper bound.
  • Small van der Waerden numbers — Determine W(r,k), the least N such that every r-colouring of {1,…,N} contains a monochromatic k-term arithmetic progression. Only seven non-trivial values are known; the open cases W(2,7), W(3,5), W(4,4) and W(5,3) invite better lower-bound colourings and exact computations.

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