A general 2-part Erd\H os-Ko-Rado theorem

Abstract

A two-part extension of the famous Erd\H{o}s-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 ${\cal F}$ 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 $|{\cal F}|$ cannot exceed the size of the largest subfamily containing one element.