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

TL;DR

This paper determines the order of magnitude of the number of maximal k-uniform intersecting families, using Tuza's set pair method to improve bounds and analyze related combinatorial functions.

Contribution

It improves existing bounds on the count of maximal intersecting families and applies Tuza's set pair approach to establish the order of magnitude of their logarithm.

Findings

Established that $M(n,k) = n^{ heta(inom{2k}{k})}$ for fixed k.

Bounded the size of the largest point set in cross-intersecting systems.

Introduced and analyzed related combinatorial functions and parameters.

Abstract

We study the function $M(n,k)$ which denotes the number of maximal $k$-uniform intersecting families $F\subseteq \binom{[n]}{k}$. 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 $k$, $M(n,k) =n^{\Theta(\binom{2k}{k})}$ 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.