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arXiv:2303.09521math.CO

An exponential improvement for diagonal Ramsey

Marcelo Campos, Simon Griffiths, Robert Morris, Julian Sahasrabudhe

Early discovery. Linked by one identifiable commenter. This paper is outside the main ranking; a mention is not an endorsement.

Abstract

The Ramsey number $R(k)$ is the minimum $n \in \mathbb{N}$ such that every red-blue colouring of the edges of the complete graph $K_n$ on $n$ vertices contains a monochromatic copy of $K_k$. We prove that \[ R(k) \leqslant (4 - \varepsilon)^k \] for some constant $\varepsilon > 0$. This is the first exponential improvement over the upper bound of Erdős and Szekeres, proved in 1935.

arXiv comments
59 pages, 8 figures
Journal reference
DOI
Subjects
math.COCombinatorics

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Linked by 1 identifiable commenter in 1 comment across 1 thread, from Dec 2023 to Dec 2023. Found in 2 HN submissions from Mar 2023 to Apr 2023. Open submission discussions.

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  1. Mathematicians have found a new upper limit to the Ramsey number 1 comment

    • I thought it was entertainingly specific, and fortunately the abstract at least [0] was slightly less immediately beyond me than I feared, to satisfy my curiosity a bit: The main proof is an improvement to the 4^k bound (standing since 1935) to (4-eps)^k for some epsilon. They…

      OJFordView on HNParent context

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