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

TL;DR

This paper characterizes the maximum size of $s$-almost $t$-intersecting families of $k$-subsets of an $n$-set, extending classical intersection theorems to a broader family of set systems.

Contribution

It provides a complete characterization of the maximum-sized $s$-almost $t$-intersecting families, generalizing the Hilton-Milner theorem.

Findings

Maximum families contain all $k$-subsets with a fixed $t$-subset.

Characterization of maximum-sized families with intersection less than $t$.

Extension of classical intersection theorems to $s$-almost $t$-intersecting families.

Abstract

A family $\mathcal{F}$ of $k$-subsets of an $n$-set is called $s$-almost $t$-intersecting if each member is $t$-disjoint with at most $s$ members. In this paper, we prove that, if $\left|\mathcal{F}\right|$ is maximum, then $\mathcal{F}$ consists of all $k$-subsets containing a fixed $t$-subset. Consequently, it is natural to consider the maximum-sized $\mathcal{F}$ with $\left|\bigcap_{F\in\mathcal{F}} F\right|<t$. The famous Hilton-Milner theorem settles the case where $\mathcal{F}$ is $t$-intersecting. We characterize the remaining case completely.