A remark on the Ramsey number of the hypercube

TL;DR

It is shown that r(Q_n) = O(2^{2n-c n}) for a universal constant $c>0$, improving upon the previous best known bound $r( Q_n)=O(2^n})$, due to Conlon, Fox and Sudakov.

Abstract

A well known conjecture of Burr and Erdos asserts that the Ramsey number $r(Q_n)$ of the hypercube $Q_n$ on $2^n$ vertices is of the order $O(2^n)$. In this paper, we show that $r(Q_n)=O(2^{2n-c n})$ for a universal constant $c>0$, improving upon the previous best known bound $r(Q_n)=O(2^{2n})$, due to Conlon, Fox and Sudakov.

Figures (1)

• Figure 1: An illustration of the "block" embedding of the cube into the bipartite graph $\Gamma$. The lower part of the vertex set of $\Gamma$ is partitioned into blocks, so that within each block every vertex has a same set of neighbors. The cube is embedded into $\Gamma$ so that each collection of vertices $\mathcal{T}_b$, $b=\pm 1$ is mapped into a single block. In the picture, the vertices of the cube which belong to $\mathcal{T}$ are enlarged, and the dotted ellipses mark the facets of the cube corresponding to the blocks of vertices. The edge density of $\Gamma$ in this illustration is made greater than $1/2$ in view of the small number of blocks (otherwise, no block embedding would be possible).

Theorems & Definitions (38)

• Theorem : CFS2016
• Theorem : CFS2016
• Theorem 1.1
• Remark
• Corollary 1.2
• Remark 1.3
• Lemma 2.1: Chernoff
• Lemma 3.1
• proof
• Corollary 3.2