On cross-2-intersecting families
Yanhong Chen, Anshui Li, Biao Wu, Huajun Zhang
TL;DR
This work determines the asymptotically sharp bound for the product of sizes of two cross-2-intersecting families A ⊆ C(n,k) and B ⊆ C(n,ℓ) under n ≥ 3.38 max{k,ℓ}, showing |A||B| ≤ C(n-2,k-2)C(n-2,ℓ-2} with equality only when both families are 2-stars around a fixed 2-subset T. The authors employ shift operators and generating-set techniques to reduce to left-compressed extremals and execute a meticulous case analysis, establishing that the trivial 2-star structure uniquely attains the bound in the stated regime; a parallel result for the k=ℓ case appears as a special case in their framework. The paper also outlines nontrivial cross-2-intersecting cases with a more intricate extremal description, providing a roadmap for obtaining related bounds via similar methods. Overall, the results advance understanding of extremal cross-intersection phenomena and broaden the toolkit for analyzing such families in high-n regimes.
Abstract
Two families $\mathcal A\subseteq\binom{[n]}{k}$ and $\mathcal B\subseteq\binom{[n]}{\ell}$ are called cross-$t$-intersecting if $|A\cap B|\geq t$ for all $A\in\mathcal A$, $B\in\mathcal B$. Let $n$, $k$ and $\ell$ be positive integers such that $n\geq 3.38\ell$ and $\ell\geq k\geq 2$. In this paper, we will determine the upper bound of $|\mathcal A||\mathcal B|$ for cross-$2$-intersecting families $\mathcal A\subseteq\binom{[n]}{k}$ and $\mathcal B\subseteq\binom{[n]}{\ell}$. The structures of the extremal families attaining the upper bound are also characterized. The similar result obtained by Tokushige can be considered as a special case of ours when $k=\ell$, but under a more strong condition $n>3.42k$. Moreover, combined with the results obtained in this paper, the complicated extremal structures attaining the upper bound for nontrivial cases can be relatively easy to reach with similar techniques.