A new measure of robustness of Erd\H{o}s--Ko--Rado Theorems on permutation groups

TL;DR

This paper introduces a novel measure for the robustness of Erd ext{"o}s--Ko--Rado theorems on permutation groups by analyzing extremal properties of Cayley subgraphs within derangement graphs.

Contribution

It proposes a new approach to quantify the robustness of EKR results using Cayley subgraphs of derangement graphs, providing new insights into permutation group properties.

Findings

New measure of robustness for EKR theorems introduced

Results on robustness for various permutation groups presented

Analysis of Cayley subgraphs enhances understanding of extremal properties

Abstract

In this paper we introduce a new way of measuring the robustness of Erd\H{o}s--Ko--Rado (EKR) Theorems on permutation groups. EKR-type results can be viewed as results about the independence numbers of certain corresponding graphs, namely the derangement graphs, and random subgraphs of these graphs have been used to measure the robustness of these extremal results. In the context of permutation groups, the derangement graphs are Cayley graphs on the permutation group in question. We propose studying extremal properties of subgraphs of derangement graphs, that are themselves Cayley graphs of the group, to measure robustness. We present a variety of results about the robustness of the EKR property of various permutation groups using this new measure.