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arXiv · 2606.17679

Typical intersecting families are trivial

Yandong Bai·Haoyun Gu·Wenston J. T. Zang
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Abstract

We study the counting problem for non-uniform intersecting families in extremal set theory. Let $J(n,k)$ denote the number of intersecting families $\mathcal{F}\subset 2^{[n]}$ such that every member of $\mathcal{F}$ has size at most $k$. Extending recent counting results for uniform intersecting families, we prove that for $n\ge 2k+2+2\sqrt{k \log k}$ and $k \rightarrow +\infty$, \[ J(n,k) =(n+o(1)) 2^{\sum_{i=1}^{k} \binom{n-1}{i-1}}. \] This result reveals that typical non-uniform intersecting families of bounded size are trivial, i.e., almost all such families share a common fixed element.

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Yandong Bai, Haoyun Gu, Wenston J. T. Zang. 2026-06-16. Typical intersecting families are trivial. https://arxiv.org/abs/2606.17679

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