The maximum sturdiness of intersecting families

TL;DR

This paper investigates the maximum sturdiness of intersecting families of sets, establishing upper bounds and optimality conditions for various types of such families in combinatorics.

Contribution

It introduces the concept of sturdiness for intersecting families and determines the maximum sturdiness for specific classes of these families, providing tight bounds.

Findings

Maximum sturdiness of intersecting families is at most binom{n-4}{k-3} for large n.

The bound is proven to be optimal.

Results apply to uniform and non-uniform intersecting families.

Abstract

Given a family $\mathcal{F}\subset 2^{[n]}$ and $1\leq i\neq j\leq n$, we use $\mathcal{F}(\bar{i},j)$ to denote the family $\{F\setminus \{j\}\colon F\in \mathcal{F},\ F\cap \{i,j\}=\{j\}\}$. The sturdiness of $\mathcal{F}$ is defined as the minimum $|\mathcal{F}(\bar{i},j)|$ over all $i,j\in [n]$ with $i\neq j$. It has a very natural algebraic definition as well. In the present paper, we consider the maximum sturdiness of $k$-uniform intersecting families, $k$-uniform $t$-intersecting families and non-uniform $t$-intersecting families. One of the main results shows that for $n\geq 36(k+6)$, an intersecting family $\mathcal{F}\subset \binom{[n]}{k}$ has sturdiness at most $\binom{n-4}{k-3}$, which is best possible.