The Hilton-Spencer cycle theorems via Katona's shadow intersection theorem

Abstract

A family A of sets is said to be intersecting if every two sets in 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 α(G) and ω(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,…,sC raised to the powers k1,…,ks, respectively, 1≤r≤α(G), and min(ω(1Ck1),…,ω(sCks))≥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(ω(1Ck1),…,ω(sCks))≥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.