Intersecting families with covering number three II

Abstract

A family $\mathcal{F}\subset \left(\genfrac{}{}{0ex}{}{[n]}{k}\right)$ is called intersecting if $F\cap F^{\prime }\ne \varnothing$ for all $F,F^{\prime }\in \mathcal{F}$. The covering number of a family $\mathcal{F}$ is defined as the minimum size of $T\subset [n]$ such that $T\cap F\ne \varnothing$ for all $F\in \mathcal{F}$. In 1980, the first author proved that for sufficiently large $n$, any intersecting $k$-graph $\mathcal{F}$ with covering number at least three, satisfies $|\mathcal{F}|\le \left(\genfrac{}{}{0ex}{}{n-1}{k-1}\right)-\left(\genfrac{}{}{0ex}{}{n-k}{k-1}\right)-\left(\genfrac{}{}{0ex}{}{n-k-1}{k-1}\right)+\left(\genfrac{}{}{0ex}{}{n-2k}{k-1}\right)+\left(\genfrac{}{}{0ex}{}{n-k-2}{k-3}\right)+3$. There was very little progress during more than forty years but recently (cf. \cite{FW25}) with a completely different approach we proved the same result for the full range $n\ge 2k$ and $k\ge 7$. In this short paper we prove the same inequality for all the remaining cases.