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Optimizing the CGMS upper bound on Ramsey numbers
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abstract
In a recent breakthrough Campos, Griffiths, Morris and Sahasrabudhe obtained the first exponential improvement of the upper bound on the diagonal Ramsey numbers since 1935. We shorten their proof, replacing the underlying book algorithm with a simple inductive statement. This modification allows us - to give a very short proof of an improved upper bound on the off-diagonal Ramsey numbers, which extends to the multicolor setting, and - to clarify the dependence of the bounds on underlying parameters and optimize these parameters, obtaining, in particular, an upper bound $$R(k,k) \leq (3.8)^{k+o(k)}$$ on the diagonal Ramsey numbers.
Forward citations
Cited by 3 Pith papers
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Ramsey numbers of trees
Every n-vertex tree with maximum degree at most cn has Ramsey number max{t1 + 2t2, 2t1} - 1, where t1 >= t2 are the sizes of its bipartition classes.
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On Ramsey-type problems for paths and cycles with few colour changes
In 3-edge-coloured complete graphs, R_3^1(P_n) equals 3n/2 + O(1) and R_3^2(C_n) equals 3n/2 + o(n) for even n.
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Upper bounds on diagonal Ramsey numbers [after Campos, Griffiths, Morris, and Sahasrabudhe]
An expository Bourbaki survey presenting the 2023 proof that r(k) ≤ (4−δ)^k via the CGMS book algorithm and the Balister et al. geometric refinement lemma, with explicit marking of every non-rigorous step.
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