An additive version of Ramsey’s theorem

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 $a, d_1, \ldots, d_k$.In particular, there is always a solution to$x_1 + \cdots+ x_\ell= y_1 + \cdots+ y_\ell$whose induced subgraph is monochromatic.

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Published 6 April 2012