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A proof of the strong Erdős--Frankl--Füredi cover-free conjecture
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)≥(r+1)2 for every integer r≥1, 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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BibTeX
@misc{hu2026proof,
title = {{A proof of the strong Erdős--Frankl--Füredi cover-free conjecture}},
author = {Yanan Hu and Leilei Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24442},
eprint = {2610.24442},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24442},
}
RIS
TY - UNPB
TI - A proof of the strong Erdős--Frankl--Füredi cover-free conjecture
AU - Yanan Hu
AU - Leilei Zhang
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24442
UR - https://arxiv.org/abs/2610.24442
AB - 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)≥(r+1)^2 for every integer r≥1, 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.
ER -