An additive version of Ramsey's theorem

TL;DR

This paper introduces a new additive variant of Ramsey's theorem, showing that for any number of colors and size, a complete graph contains a monochromatic subgraph with vertices defined by additive combinations, extending classical Ramsey results.

Contribution

The paper presents a novel additive formulation of Ramsey's theorem, demonstrating the existence of monochromatic subgraphs with vertices expressed as sums of subsets, which is a new perspective in combinatorial Ramsey theory.

Findings

Existence of monochromatic subgraphs with vertices as additive sums

Generalization of classical Ramsey's theorem to additive structures

Applicable for any number of colors and subgraph size

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.