The maximum sturdiness of intersecting families

Abstract

Given a family $\mathcal{F}\subset 2^{[n]}$ and $1\le i\ne j\le n$, we use $\mathcal{F}(\bar{i},j)$ to denote the family $\{F\setminus \{j\}:F\in \mathcal{F},F\cap \{i,j\}=\{j\}\}$. The sturdiness of $\mathcal{F}$ is defined as the minimum $|\mathcal{F}(\bar{i},j)|$ over all $i,j\in [n]$ with $i\ne j$. 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 $n\ge 36(k+6)$, an intersecting family $\mathcal{F}\subset \left(\genfrac{}{}{0ex}{}{[n]}{k}\right)$ has sturdiness at most $\left(\genfrac{}{}{0ex}{}{n-4}{k-3}\right)$, which is best possible.

19 pages