On the Hilton-Spencer intersection theorems for unions of cycles

TL;DR

This paper provides a concise proof of Hilton-Spencer intersection theorems for unions of cycles, establishing conditions under which the largest intersecting families are trivial, using Katona's Shadow Intersection Theorem.

Contribution

It offers a simplified proof for Hilton-Spencer intersection theorems on unions of cycles, extending previous results with relaxed clique number conditions.

Findings

Proves the Hilton-Spencer intersection theorems using Katona's Shadow Intersection Theorem.

Establishes conditions for trivial maximum intersecting families in unions of cycles.

Provides a shorter proof for cases with strict inequalities on clique numbers.

Abstract

A family $\mathcal{A}$ of sets is said to be intersecting if every two sets in $\mathcal{A}$ intersect. An intersecting family is said to be \emph{trivial} it its sets have a common element. A graph $G$ is said to be $r$-EKR if at least one of the largest intersecting families of independent $r$-element sets of $G$ is trivial. Let $\alpha(G)$ and $\omega(G)$ denote the independence number and the clique number of $G$, respectively. Hilton and Spencer recently showed that if $G$ is the vertex-disjoint union of a cycle ${_*C}$ raised to the power $k^*$ and $s$ cycles ${_1C}, \dots, {_sC}$ raised to the powers $k_1, \dots, k_s$, respectively, $1 \leq r \leq \alpha(G)$, and $$\min\big(\omega(_1C^{k_1}), \dots, \omega(_sC^{k_s})\big) \geq 2k^* + 1,$$ then $G$ is $r$-EKR. They had shown that the same holds if ${_*C}$ is replaced by a path and the condition on the clique numbers is relaxed to $$\min\big(\omega(_1C^{k_1}), \dots, \omega(_sC^{k_s})\big) \geq k^* + 1.$$ We use the classical Shadow Intersection Theorem of Katona to obtain a short proof of each result for the case where the inequality for the minimum clique number is strict.