Intersecting families with covering number three

Abstract

We consider $k$-graphs on $n$ vertices, that is, $\mathcal{F}\subset \left(\genfrac{}{}{0ex}{}{[n]}{k}\right)$. A $k$-graph $\mathcal{F}$ is called intersecting if $F\cap F^{\prime }\ne \varnothing$ for all $F,F^{\prime }\in \mathcal{F}$. In the present paper we prove that for $k\ge 7$, $n\ge 2k$, any intersecting $k$-graph $\mathcal{F}$ with covering number at least three, satisfies $|\mathcal{F}|\le \left(\genfrac{}{}{0ex}{}{n-1}{k-1}\right)-\left(\genfrac{}{}{0ex}{}{n-k}{k-1}\right)-\left(\genfrac{}{}{0ex}{}{n-k-1}{k-1}\right)+\left(\genfrac{}{}{0ex}{}{n-2k}{k-1}\right)+\left(\genfrac{}{}{0ex}{}{n-k-2}{k-3}\right)+3$, the best possible upper bound which was proved in \cite{F80} subject to exponential constraints $n>n_0(k)$.

31 pages