A general 2-part Erdȍs-Ko-Rado theorem

Abstract

A two-part extension of the famous Erdȍs-Ko-Rado Theorem is proved. The underlying set is partitioned into \(X_1\) and \(X_2\). Some positive integers \(k_i\), \(\ell_i\) (\(1\leq i\leq m\)) are given. We prove that if \(\mathcal{F}\)) is an intersecting family containing members \(F\) such that \(|F\cap X_1|=k_i\), \(|F\cap X_2|=\ell_i\) holds for one of the values \(i\) (\(1\leq i\leq m\)) then \(|\mathcal{F}|\) cannot exceed the size of the largest subfamily containing one element.

Keywords

• extremal set theory
• two-part problem
• intersecting family