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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 $\mathcal{Q}_d$, the $d$-dimensional hypercube, Ramsey if any 2-coloring of the vertices of $\mathcal{Q}_n$, for $n$ sufficiently large, contains a monochromatic copy of $S$. Ramsey's theorem tells us that for any $r\geq1$ 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 $\mathcal{Q}_d$ 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 $\mathcal{Q}_d$ are Ramsey? The answer to this question depends on the number of cliques involved. In particular we determine which unions of two or three cliques are Ramsey and then show, using a probabilistic argument, that any nontrivial union of 39 or more cliques of different weights cannot be Ramsey. A key tool is a lemm...

Vertex Ramsey Problems in the Hypercube