An Erds-Ko-Rado Type Theorem via the Polynomial Method

Abstract

A family F is an intersecting family if any two members have a nonempty intersection. Erdos, Ko, and Rado showed that vertical bar F vertical bar <= (n(k 1)(n-1)) holds for a k-uniform intersecting family F of subsets of [n]. The Erdos-Ko-Rado theorem for non-uniform intersecting families of subsets of [n] of size at most k can be easily proved by applying the above result to each uniform subfamily of a given family. It establishes that vertical bar F vertical bar <= ((n-1)(k-1)) + ((n - 1)(k - 2)) + . . . + (n(0)(n) (-) (1)) holds for non-uniform intersecting families of subsets of [n] of size at most k. In this paper, we prove that the same upper bound of the Erdos-Ko-Rado Theorem for k-uniform intersecting families of subsets of [n] holds also in the non-uniform family of subsets of [n] of size at least k and at most n k with one more additional intersection condition. Our proof is based on the method of linearly independent polynomials.