Erd\H{o}s-Ko-Rado theorem in Peisert-type graphs

TL;DR

This paper extends the Erd ext{"o}s-Ko-Rado theorem to almost all pseudo-Paley graphs of Peisert-type, demonstrating that maximum cliques are canonical in these algebraically structured graphs and establishing related stability results.

Contribution

It generalizes the EKR theorem to a broad class of Peisert-type graphs, including pseudo-Paley graphs, and proves stability results for maximum cliques.

Findings

EKR theorem holds for almost all pseudo-Paley graphs of Peisert-type.

Maximum cliques are canonical in these graphs.

Stability results for maximum cliques are established.

Abstract

The celebrated Erd\H{o}s-Ko-Rado (EKR) theorem for Paley graphs (of square order) states that all maximum cliques are canonical in the sense that each maximum clique arises from the subfield construction. Recently, Asgarli and Yip extended this result to Peisert graphs and other Cayley graphs which are Peisert-type graphs with nice algebraic properties on the connection set. On the other hand, there are Peisert-type graphs for which the EKR theorem fails to hold. In this paper, we show that the EKR theorem of Paley graphs extends to almost all pseudo-Paley graphs of Peisert-type. Furthermore, we establish the stability results of the same flavor.