Ramsey Numbers in Kneser Graphs

TL;DR

This paper investigates the minimum size of Kneser graphs needed to guarantee monochromatic cliques under two-colorings, providing bounds and advancing understanding of related Ramsey problems in combinatorics.

Contribution

It establishes general bounds on r-Kneser Ramsey numbers and advances two open problems in the field of combinatorial Ramsey theory.

Findings

Derived bounds for r-Kneser Ramsey numbers.

Made progress on two open Ramsey-type problems.

Enhanced understanding of monochromatic clique existence in Kneser graphs.

Abstract

We define the $r\textit{-Kneser Ramsey number}$ $R^{\textrm{KG}}_{r}(s, t)$ as the minimum integer $n$ such that every red/blue edge-coloring of the Kneser graph $\textrm{KG}(n,r)$ contains a red $s$-clique or a blue $t$-clique. We obtain general bounds on the numbers $R^{\textrm{KG}}_{r}(s, t)$, and make progress on two related Ramsey-type problems, one raised by Holmsen, Hrusak, and Rold\'an-Pensado, and the other posted by P\'alv\"olgyi.