Intersecting families with covering number three

Abstract

We consider k-graphs on n vertices, that is, $F\subset \left(\begin{array}{c}\left[n\right]\\ k\end{array}\right)$. A k-graph $F$ is called intersecting if $F\cap F^{\prime }\ne \varnothing$ for all $F,F^{\prime }\in F$. In the present paper we prove that for $k\ge 7$, $n\ge 2k$, any intersecting k-graph $F$ with covering number at least three, satisfies $\left|F\right|\le \left(\begin{array}{c}n-1\\ k-1\end{array}\right)-\left(\begin{array}{c}n-k\\ k-1\end{array}\right)-\left(\begin{array}{c}n-k-1\\ k-1\end{array}\right)+\left(\begin{array}{c}n-2k\\ k-1\end{array}\right)+\left(\begin{array}{c}n-k-2\\ k-3\end{array}\right)+3$, the best possible upper bound which was proved in [4] subject to exponential constraints $n>n_0\left(k\right)$.