A proof of the strong Erdős--Frankl--Füredi cover-free conjecture

Yanan Hu; Leilei Zhang

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arXiv · 2610.24442

A proof of the strong Erdős--Frankl--Füredi cover-free conjecture

Yanan Hu·Leilei Zhang

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Abstract

A family of nonempty sets is $r$-cover-free if no member is contained in the union of at most $r$ other members. Let $T(r)$ denote the least $n$ for which there is an $r$-cover-free family on $n$ points with more than $n$ members. In this paper, we prove that $T(r)\ge(r+1)^2$ for every integer $r\ge1$, confirming the strong form of the Erdős--Frankl--Füredi cover-free conjecture. In particular, $T(r)=(r+1)^2$ whenever $r+1$ is a prime power.

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Yanan Hu, Leilei Zhang. 2026-10-17. A proof of the strong Erdős--Frankl--Füredi cover-free conjecture. https://arxiv.org/abs/2610.24442

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