On the Ramsey number of the triangle and the cube

Abstract

The Ramsey number r(K 3,Q n ) is the smallest integer N such that every red-blue colouring of the edges of the complete graph K N contains either a red n-dimensional hypercube, or a blue triangle. Almost thirty years ago, Burr and Erdős conjectured that r(K 3,Q n )=2n+1−1 for every n∈ℕ, but the first non-trivial upper bound was obtained only recently, by Conlon, Fox, Lee and Sudakov, who proved that r(K 3,Q n )⩽7000·2n. Here we show that r(K 3,Q n )=(1+o(1))2n+1 as n→∞.

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Correspondence to Jozef Skokan.

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Research supported in part by: CNPq bolsas PDJ (GFP, SG, DS), a CNPq bolsa de Produtividade em Pesquisa (RM). This work was carried out during a visit of JS to IMPA in November 2012

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Pontiveros, G.F., Griffiths, S., Morris, R. et al. On the Ramsey number of the triangle and the cube. Combinatorica 36, 71–89 (2016). https://doi.org/10.1007/s00493-015-3089-8

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Mathematics Subject Classification (2000)

  • 05C55