Ramsey Numbers in Kneser Graphs

TL;DR

The paper introduces and studies the r-Kneser Ramsey numbers $R^{\textrm{KG}}_{r}(s,t)$ for Kneser graphs, establishing a general framework that recovers classical Ramsey numbers when $r=1$ and extends Ramsey theory to disjoint $r$-subsets. It develops both universal upper and lower bounds, using partition-based cliques in Kneser graphs and AFL-type theorems, and provides concrete computations for small parameters via SAT-based methods. The authors make progress on two Ramsey-type problems: improving the beta threshold in the Holmsen–Hrusak–Roldán-Pensado problem to $7/3$ and obtaining positive results for $\alpha<1/4$ and $t=3$ (with a computational counterexample at $\alpha=1/4$). They also study induced Kneser graphs in Palvolgyi’s problem, showing a general lower bound $N \ge R_r(3r,3r)$ and constructing colorings that avoid red induced $\textrm{KG}(3r,r)$ and blue triangles, thereby advancing understanding of induced-structure Ramsey phenomena on KG graphs. Overall, the work blends combinatorial constructions, topological bounds, and computational searches to map the landscape of Kneser Ramsey numbers and their Ramsey-type variants.

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 Roldán-Pensado, and the other posted by Pálvölgyi.

Figures (1)

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Theorems & Definitions (25)

• Definition 1
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Theorem 9
• Proposition 10
• Theorem 11
• Theorem 12
• Theorem 14