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.