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arXiv · 2412.07090

The maximum sturdiness of intersecting families

Peter Frankl·Jian Wang
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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.

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Peter Frankl, Jian Wang. 2024-12-10. The maximum sturdiness of intersecting families. https://arxiv.org/abs/2412.07090

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