Vertex Ramsey problems in the hypercube

TL;DR

This paper investigates which vertex sets in high-dimensional hypercubes are guaranteed to appear monochromatic in any 2-coloring, identifying specific unions of cliques that are Ramsey and those that are not.

Contribution

It characterizes unions of 2 or 3 cliques as Ramsey and proves that unions of 39 or more cliques are not, using probabilistic methods and a key lemma relating hypercube configurations to integer sets.

Findings

Unions of 2 or 3 cliques are Ramsey.

Unions of 39 or more cliques are not Ramsey.

A lemma connects hypercube problems to integer translate problems.

Abstract

If we 2-color the vertices of a large hypercube what monochromatic substructures are we guaranteed to find? Call a set S of vertices from Q_d, the d-dimensional hypercube, Ramsey if any 2-coloring of the vertices of Q_n, for n sufficiently large, contains a monochromatic copy of S. Ramsey's theorem tells us that for any r \geq 1 every 2-coloring of a sufficiently large r-uniform hypergraph will contain a large monochromatic clique (a complete subhypergraph): hence any set of vertices from Q_d that all have the same weight is Ramsey. A natural question to ask is: which sets S corresponding to unions of cliques of different weights from Q_d are Ramsey? The answer to this question depends on the number of cliques involved. In particular we determine which unions of 2 or 3 cliques are Ramsey and then show, using a probabilistic argument, that any non-trivial union of 39 or more cliques of different weights cannot be Ramsey. A key tool is a lemma which reduces questions concerning monochromatic configurations in the hypercube to questions about monochromatic translates of sets of integers.