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

Dehai Liu, Kaishun Wang, Tian Yao

Published 2024-10-26Version 1

Let $\mathcal{F}$ be a family of $k$-subsets of an $n$-set. Write $\mathcal{D}_{\mathcal{F}}(H;t)=\left\{ F\in\mathcal{F}: \left|F\cap H\right|<t\right\}$ for a set $H$. The family $\mathcal{F}$ is called $s$-almost $t$-intersecting if $\left|\mathcal{D}_{\mathcal{F}}(F;t)\right|\leq s$ for each $F\in \mathcal{F}$. In this paper, we prove that $s$-almost $t$-intersecting families with maximum size are $t$-intersecting. We also consider $s$-almost $t$-intersecting families which are not $t$-intersecting, and characterize such families with maximum size.