The Ramsey number R(5,5)

Determine R(5,5), now known to lie between 43 and 46: colour K_43 with no one-coloured K_5, or prove that 45 vertices force one.

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State the problem, from Small Ramsey Numbers (Radziszowski, dynamic survey DS1)

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	1	+# The Ramsey number R(5,5)
	2	+
	3	+What is the least $n$ such that every red-blue colouring of the edges of the complete graph $K_n$ contains five vertices joined only by edges of one colour? This number, $R(5,5)$, is known to lie between 43 and 46. A colouring of $K_{43}$ with no single-coloured $K_5$ would be checked in seconds; a better upper bound needs a large but repeatable computation.
	4	+
	5	+## Problem
	6	+
	7	+The Ramsey number $R(k, l)$ is the least $n$ such that every graph on $n$ vertices contains a clique of size $k$ or an independent set of size $l$; equivalently, every 2-colouring of the edges of $K_n$ contains a red $K_k$ or a blue $K_l$. Ramsey's theorem guarantees that these numbers are finite.
	8	+
	9	+Task: determine $R(5,5)$. It is known that $43 \le R(5,5) \le 46$.
	10	+
	11	+## Background
	12	+
	13	+Exact values are known for only a few small parameters: $R(3,3) = 6$ is a classical exercise, Greenwood and Gleason found $R(3,4) = 9$, $R(3,5) = 14$ and $R(4,4) = 18$ in 1955, and McKay and Radziszowski proved $R(4,5) = 25$ in 1995. $R(5,5)$ is the smallest open diagonal case. Lower bounds come from explicit colourings, usually found by computer search; upper bounds require ruling out every colouring of a given size, which for $R(5,5)$ has meant combining counting identities, linear programming and very large case analyses. A value of 50 that circulates online is wrong, as the survey notes.
	14	+
	15	+## What is known
	16	+
	17	+- Lower bound $R(5,5) \ge 43$: Exoo (1989) gave a 2-colouring of $K_{42}$ with no monochromatic $K_5$.
	18	+- Upper bounds: $49$ (McKay and Radziszowski, 1997); $48$ (Angeltveit and McKay, 2018); $46$ (Angeltveit and McKay, posted 2024, published in the Journal of Graph Theory), by linear programming together with a large number of cases checked by computer, each computation implemented independently by both authors.
	19	+- Conjecture: McKay and Radziszowski (1997) gave strong evidence that $R(5,5) = 43$ and that there are exactly 656 colourings of $K_{42}$ without a monochromatic $K_5$, up to isomorphism.
	20	+
	21	+## What would count as progress
	22	+
	23	+- A 2-colouring of $K_{43}$ with no monochromatic $K_5$, which would show $R(5,5) \ge 44$ and refute the conjecture. Checking it means testing the $\binom{43}{5} = 962{,}598$ five-vertex subsets, a matter of seconds.
	24	+- A proof that $R(5,5) \le 45$, with the linear programs and case enumerations published so that they can be rerun independently.
	25	+- New structural information about extremal colourings, for example a complete enumeration of the colourings of $K_{42}$ without a monochromatic $K_5$.
	26	+- The problem is stated in Lean in [formal-conjectures](https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Wikipedia/RamseyNumbers.lean), with the bounds $43 \le R(5,5) \le 46$.
	27	+
	28	+## References
	29	+
	30	+1. G. Exoo, "A lower bound for R(5,5)", Journal of Graph Theory 13 (1989), 97–98. [doi:10.1002/jgt.3190130113](https://doi.org/10.1002/jgt.3190130113)
	31	+2. B. D. McKay and S. P. Radziszowski, "Subgraph counting identities and Ramsey numbers", Journal of Combinatorial Theory, Series B 69 (1997), 193–209. [doi:10.1006/jctb.1996.1741](https://doi.org/10.1006/jctb.1996.1741)
	32	+3. V. Angeltveit and B. D. McKay, "R(5,5) ≤ 48", Journal of Graph Theory 89 (2018), 5–13. [arXiv:1703.08768](https://arxiv.org/abs/1703.08768)
	33	+4. V. Angeltveit and B. D. McKay, "R(5,5) ≤ 46" (2024; Journal of Graph Theory, 2026). [arXiv:2409.15709](https://arxiv.org/abs/2409.15709)
	34	+5. S. P. Radziszowski, "Small Ramsey numbers", Electronic Journal of Combinatorics, Dynamic Survey DS1, revision 18 (2026). [doi:10.37236/21](https://doi.org/10.37236/21)
	35	+
	36	+## Source
	37	+
	38	+Imported from [Small Ramsey Numbers (Radziszowski, dynamic survey DS1), Table Ia](https://www.combinatorics.org/ds1), curated by Stanisław Radziszowski. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.