Almost Intersecting Families

Abstract

Let $n > k > 1$ be integers, $[n] = \{1, \ldots, n\}$. Let $\mathcal F$ be a family of $k$-subsets of $[n]$. The family $\mathcal F$ is called intersecting if $F \cap F' \neq \varnothing$ for all $F, F' \in \mathcal F$. It is called almost intersecting if it is not intersecting but to every $F \in \mathcal F$ there is at most one $F'\in \mathcal F$ satisfying $F \cap F' = \varnothing$. Gerbner et al. proved that if $n \geq 2k + 2$ then $|\mathcal F| \leqslant {n - 1\choose k - 1}$ holds for almost  intersecting families. Our main result implies the considerably stronger and best possible bound $|\mathcal F| \leqslant {n - 1\choose k - 1} - {n - k - 1\choose k - 1} + 2$ for $n > (2 + o(1))k$, $k\ge 3$.