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.
From: Yandong Bai [view email]
[v1]
Tue, 16 Jun 2026 08:46:20 UTC (7 KB)
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