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

Abstract

A two-part extension of the famous Erdős-Ko-Rado Theorem is proved. The underlying set is partitioned into $X_1$ and $X_2$. Some positive integers $k_i,{\ell }_i(1\le i\le m)$ 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 $i(1\le i\le m)$ then $|\mathcal{F}|$ cannot exceed the size of the largest subfamily containing one element.