Uniform intersecting families with large covering number

Peter Frankl, Andrey Kupavskii

A family $\mathcal F$ has covering number $\tau$ if the size of the smallest set intersecting all sets from $\mathcal F$ is equal to $s$. Let $m(n,k,\tau)$ stand for the size of the largest intersecting family $\mathcal F$ of $k$-element subsets of $\{1,\ldots,n\}$ with covering number $\tau$. It is a classical result of Erd\H os and Lov\'asz that $m(n,k,k)\le k^k$ for any $n$. In this short note, we explore the behaviour of $m(n,k,\tau)$ for $nk-\frac 12k^{1/2}$.

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