Intersecting families, cross-intersecting families, and a proof of a conjecture of Feghali, Johnson and Thomas

摘要

A family A of sets is said to be intersecting if every two sets in A intersect. Two families A and B are said to be cross-intersecting if each set in A intersects each set in B. For a positive integer n, let [n] {1, ... , n} and S-n = {A subset of [n] : 1 is an element of A}. We extend the Erdoss-Ko-Rado Theorem by showing that if A and B are non-empty cross-intersecting families of subsets of [n], A is intersecting, and a(0), a(1), ... , a(n), b(0), b(1), ... , b(n) are non-negative real numbers such that a(i) + b(i) = + a(n-i) + b(n-i) and a(n-i) = b(i) for each i = n/2, then