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

TL;DR

This paper extends a classical combinatorial theorem by establishing an upper bound on the product of sizes of families with a weaker intersection condition, generalizing Pyber's theorem for large n.

Contribution

It introduces a new intersection condition involving sums of intersections and proves an upper bound on the product of family sizes, generalizing previous results.

Findings

Established an upper bound for the product of family sizes under a new intersection condition.

Extended Pyber's theorem to a broader class of cross-intersecting families.

Proved the result for sufficiently large n.

Abstract

We prove that if two families $\mathcal{F} \subseteq \binom{[n]}{k}$ and $\mathcal{F}' \subseteq \binom{[n]}{k'}$ satisfy $\sum_{1 \leq i, j \leq \ell} \lvert F_i \cap F_j' \rvert \geq \ell^2t - \ell +1$ for every choice of distinct $F_1, \ldots, F_\ell \in \mathcal{F}$ and $F_1', \ldots, F_\ell' \in \mathcal{F}'$, then $\lvert \mathcal{F} \rvert \cdot \lvert \mathcal{F}' \rvert \leq \binom{n-t}{k-t} \binom{n-t}{k'-t}$, 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.