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A new measure of robustness of Erdős--Ko--Rado Theorems on permutation groups

Karen Gunderson, Karen Meagher, Joy Morris, Venkata Raghu Tej Pantangi, Mahsa N Shirazi

TL;DR

This work introduces a quantitative notion of robustness for Erdős--Ko--Rado phenomena on permutation groups by examining independence numbers of Cayley subgraphs $\mathrm{Cay}(G,\mathrm{Der}(G)\setminus D)$ obtained from the derangement graph $\Gamma_G=\mathrm{Cay}(G,\mathrm{Der}(G))$ after removing inverse-closed label sets $D$. It defines the critical parameter $\mathbf{d}_G$ as the minimal number of label-pairs whose removal can alter the independence number and demonstrates robustness for several natural group actions (e.g., regular, Frobenius, generalized dihedral) while detailing how independence numbers can jump to multiples like $2\alpha(\Gamma_G)$ (or $3\alpha$ when order-3 elements occur) in dihedral settings. The paper provides precise classifications for one- and two-label deletions in dihedral cases, shows strong robustness for $\mathrm{PGL}(2,q)$, and analyzes the more nuanced behavior in $\mathrm{Sym}(n)$, including $\mathrm{Sym}(5)$ not being robust. It also develops subgroup- and homomorphism-based tools to transfer robustness results and ends with open problems, including questions about robustness for larger symmetric/alternating groups and other Cayley-graph families.

Abstract

In this paper we introduce a new way of measuring the robustness of Erdős--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.

A new measure of robustness of Erdős--Ko--Rado Theorems on permutation groups

TL;DR

This work introduces a quantitative notion of robustness for Erdős--Ko--Rado phenomena on permutation groups by examining independence numbers of Cayley subgraphs obtained from the derangement graph after removing inverse-closed label sets . It defines the critical parameter as the minimal number of label-pairs whose removal can alter the independence number and demonstrates robustness for several natural group actions (e.g., regular, Frobenius, generalized dihedral) while detailing how independence numbers can jump to multiples like (or when order-3 elements occur) in dihedral settings. The paper provides precise classifications for one- and two-label deletions in dihedral cases, shows strong robustness for , and analyzes the more nuanced behavior in , including not being robust. It also develops subgroup- and homomorphism-based tools to transfer robustness results and ends with open problems, including questions about robustness for larger symmetric/alternating groups and other Cayley-graph families.

Abstract

In this paper we introduce a new way of measuring the robustness of Erdős--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.

Paper Structure

This paper contains 14 sections, 34 theorems, 36 equations, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Cliques in $\Gamma_G$ from subgroups of $G$
  4. Dihedral and Generalised Dihedral Groups
  5. Removal a single label from connection set
  6. Removal of two labels from Dihedral groups
  7. Robustness in generalised dihedral groups
  8. Generalised dihedral groups over subgroups of the alternating group
  9. $\mathrm{PGL}(2,q)$ and symmetric groups
  10. $\mathrm{PGL}(2,q)$
  11. Symmetric groups
  12. Subgroups and Homomorphisms
  13. Subgroups with a unique derangement
  14. Open problems and future work

Key Result

Proposition 2.1

Let $i, j$ be distinct fixed elements in $[n]$ and let $D_{i\to j}$ be the set of all derangements in a transitive group $G \leq \mathrm{Sym}({n})$ that map $i$ to $j$. Consider $\mathrm{Cay}(G, \mathrm{Der}(G) \setminus D )$, where $D = D_{i\to j} \cup D_{j \to i}$ (so all the permutations that map

Theorems & Definitions (67)

  • Definition 1.1
  • Example 1
  • Proposition 2.1
  • proof
  • Theorem 2.2: Clique-coclique bound
  • Proposition 2.3
  • proof
  • Corollary 2.4
  • proof
  • Proposition 2.5
  • ...and 57 more