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 trivial if 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 1 C,. .. , s C raised to the powers k 1 ,. .. , k s , respectively, 1 ≤ r ≤ α(G), and min ω 1 C k1 ,. .. , ω s C ks ≥ ω C k , then G is r-EKR. They had shown that the same holds if C is replaced by a path P and the condition on the clique numbers is relaxed to min ω 1 C k1 ,. .. , ω s C ks ≥ ω P k. We use the classical Shadow Intersection Theorem of Katona to obtain a significantly shorter proof of each result for the case where the inequality for the minimum clique number is strict.