THE HILTON-SPENCER CYCLE THEOREMS VIA KATONA'S SHADOW INTERSECTION THEOREM

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 a(G) and ?(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,... , 3C raised to the powers k1, ... , k3, respectively, 1 < r < a(G), andmin (? (C-1(k1)), ... , ?(C-s(s)k)) = ?(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 tomin (? (1Ck1), ..., ? ( 3Cks)) & GE; ?(Pk).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.