INTERSECTING FAMILIES IN SYMMETRIC UNIONS OF DIRECT PRODUCTS OF SET FAMILIES

摘要

Let X-1,...,X-m be m pairwise disjoint sets with the same size n, and let {k(1),...,k(m)} be a multiset of positive integers such that k(i) = n/2. Let X = X-1 boolean OR ... boolean OR X-m and k = k(1) +...+ k(m). Given a k-set A subset of X, let I(A) be the m-element multiset {vertical bar A boolean AND X-1 vertical bar,..., vertical bar A boolean AND X-m vertical bar}. Set Omega = {A is an element of ((X)(k)) : I(A) = {k(1),...,k(m)}}. In this paper, we prove that if F is an intersecting family in Omega, then vertical bar F vertical bar = kmp)(k(1),..., k(m)) Pi(m)(i=1) ((n)(ki)), where p(k(1),...,k(m)) equals the number of permutations of k(1),...,k(m). Furthermore, equality holds if and only if F = {A is an element of Omega : a is an element of A} for some a is an element of X, except in the case k(1) = ... = k(m) = n/2.