On the number of maximal intersecting k-uniform families and further applications of Tuza's set pair method

Abstract

We study the function $M(n,k)$ which denotes the number of maximal $k$-uniform intersecting families $F\subseteq \left(\genfrac{}{}{0ex}{}{[n]}{k}\right)$. Improving a bound of Balogh at al. on $M(n,k)$, we determine the order of magnitude of $\log M(n,k)$ by proving that for any fixed $k$, $M(n,k)=n^{\Theta ((\genfrac{}{}{0ex}{}{2k}{k}))}$ holds. Our proof is based on Tuza's set pair approach. The main idea is to bound the size of the largest possible point set of a cross-intersecting system. We also introduce and investigate some related functions and parameters.

11 pages