$s$-almost cross-$t$-intersecting families for finite sets

TL;DR

This paper characterizes the largest possible product of sizes for $s$-almost cross-$t$-intersecting families of $k$-subsets, extending understanding of near-intersecting set families with stability results.

Contribution

It introduces a characterization of maximum-sized $s$-almost cross-$t$-intersecting families and establishes stability results for near-intersecting families.

Findings

Maximum product of sizes characterized

Stability results for near-intersecting families established

Extension of intersecting family theory to $s$-almost cases

Abstract

Two families $\mathcal{F}$ and $\mathcal{G}$ of $k$-subsets of an $n$-set are called $s$-almost cross-$t$-intersecting if each member in $\mathcal{F}$ (resp. $\mathcal{G}$) is $t$-disjoint with at most $s$ members in $\mathcal{G}$ (resp. $\mathcal{F}$). In this paper, we characterize the $s$-almost cross-$t$-intersecting families with the maximum product of their sizes. Furthermore, we provide a corresponding stability result after studying the $s$-almost cross-$t$-intersecting families which are not cross-$t$-intersecting.