On a weaker notion of cross $t$-intersecting families

Abstract

We prove that if two families $\mathcal{F}\subseteq \left(\genfrac{}{}{0ex}{}{[n]}{k}\right)$ and ${\mathcal{F}}^{\prime }\subseteq \left(\genfrac{}{}{0ex}{}{[n]}{k^{\prime }}\right)$ satisfy ${\sum }_{1\le i,j\le \ell }|F_i\cap F_j^{\prime }|\ge {\ell }^{2}t-\ell +1$ for every choice of distinct $F_1,\dots ,F_{\ell }\in \mathcal{F}$ and $F_1^{\prime },\dots ,F_{\ell }^{\prime }\in {\mathcal{F}}^{\prime }$, then $|\mathcal{F}|\cdot |{\mathcal{F}}^{\prime }|\le \left(\genfrac{}{}{0ex}{}{n-t}{k-t}\right)\left(\genfrac{}{}{0ex}{}{n-t}{k^{\prime }-t}\right)$, provided that $n$ is sufficiently large. This extends a celebrated theorem of Pyber for large $n$, which determines the tight upper bound for the product of the sizes of cross $1$-intersecting families.

5 pages