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arXiv · 2611.06860

Largest Sperner families with bounded VC-dimension

Gennian Ge·Hanyu Wang·Zixiang Xu·Yixuan Zhang
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Abstract

A conjecture of Frankl proposed in the 1980s states that for $n\ge2d$, every Sperner family $\mathcal{F}\subseteq 2^{[n]}$ of VC-dimension at most $d$ has at most $\binom{n}{d}$ members. We prove this conjecture for every fixed $d$ and large $n$, and show that equality holds only when $\mathcal{F}$ is $\binom{[n]}{d}$ or $\binom{[n]}{n-d}$.

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Gennian Ge, Hanyu Wang, Zixiang Xu, Yixuan Zhang. 2026-11-04. Largest Sperner families with bounded VC-dimension. https://arxiv.org/abs/2611.06860

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