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.