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arXiv · 2601.20516

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

Jiangdong Ai·Ming Chen·Seokbeom Kim·Hyunwoo Lee
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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.

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Jiangdong Ai, Ming Chen, Seokbeom Kim, Hyunwoo Lee. 2026-01-28. On a weaker notion of cross $t$-intersecting families. https://arxiv.org/abs/2601.20516

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