On the use of senders for asymmetric tuples of cliques in Ramsey theory

TL;DR

This paper constructs signal sender gadgets for asymmetric tuples of cliques in multicolor Ramsey theory and uses them to extend classical theorems to this more general setting.

Contribution

It introduces the first known gadgets for asymmetric multicolor tuples of cliques and applies them to generalize key Ramsey theorems.

Findings

Constructed signal senders for any tuple of cliques in multicolor Ramsey theory.

Generalized classical theorems to the asymmetric multicolor setting.

Enhanced understanding of graph properties in multicolor Ramsey problems.

Abstract

A graph $G$ is $q$-Ramsey for a $q$-tuple of graphs $(H_1,\ldots,H_q)$ if for every $q$-coloring of the edges of $G$ there exists a monochromatic copy of $H_i$ in color $i$ for some $i\in[q]$. Over the last few decades, researchers have investigated a number of questions related to this notion, aiming to understand the properties of graphs that are $q$-Ramsey for a fixed tuple. Among the tools developed while studying questions of this type are gadget graphs, called signal senders and determiners, which have proven invaluable for building Ramsey graphs with certain properties. However, until now these gadgets have been shown to exist and used mainly in the two-color setting or in the symmetric multicolor setting, and our knowledge about their existence for multicolor asymmetric tuples is extremely limited. In this paper, we construct such gadgets for any tuple of cliques. We then use these gadgets to generalize three classical theorems in this area to the asymmetric multicolor setting.