Ramsey Numbers in Kneser Graphs

Abstract

We define the $r\text{-Kneser Ramsey number}$ $R_r^{\text{KG}}(s,t)$ as the minimum integer $n$ such that every red/blue edge-coloring of the Kneser graph $\text{KG}(n,r)$ contains a red $s$-clique or a blue $t$-clique. We obtain general bounds on the numbers $R_r^{\text{KG}}(s,t)$, and make progress on two related Ramsey-type problems, one raised by Holmsen, Hrusak, and Roldán-Pensado, and the other posted by Pálvölgyi.