An additive version of Ramsey's theorem

Abstract

We show that, for every $r, k$, there is an $n = n(r,k)$ so that any $r$-coloring of the edges of the complete graph on $[n]$ will yield a monochromatic complete subgraph on vertices ${a + \sum_{i \in I} d_i \mid I \subseteq [k]}$ for some choice of $a, d_1,..., d_k$. In particular, there is always a solution to $x_1 + ... + x_\ell = y_1 + ... + y_\ell$ whose induced subgraph is monochromatic.