On the Ramsey number of the triangle and the cube
Combinatorics
2013-02-18 v1
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 Erd\H{o}s conjectured that r(K_3,Q_n) = 2^{n+1} - 1 for every n \in \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) \le 7000 \cdot 2^n. Here we show that r(K_3,Q_n) = (1 + o(1)) 2^{n+1} as n \to \infty.
Cite
@article{arxiv.1302.3840,
title = {On the Ramsey number of the triangle and the cube},
author = {Gonzalo Fiz Pontiveros and Simon Griffiths and Robert Morris and David Saxton and Jozef Skokan},
journal= {arXiv preprint arXiv:1302.3840},
year = {2013}
}
Comments
16 pages