PreprintCC BY 4.0arXiv
The second-order term for the largest r-fork-free families
Abstract
A family of subsets of [n] is r-fork-free if none of its members is strictly contained in r other distinct members. For each fixed integer r≥2, we prove that the maximum size of such a family is binomn⌊ n/2⌋ (1+(2(r−1))/n+o(n−1)). This determines the second-order term and matches the upper bound of De Bonis and Katona. Our lower bound comes from a construction on two adjacent middle levels, using a finite ordered collection of disjoint coordinate blocks and a condition on sums modulo n. For each prescribed error, the blocks are fixed before n tends to infinity, so the construction works for every sufficiently large n.
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BibTeX
@misc{zhan2026secondorder,
title = {{The second-order term for the largest $r$-fork-free families}},
author = {Yiyan Zhan and Mei Lu and Xiamiao Zhao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30336},
eprint = {2609.30336},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30336},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The second-order term for the largest r-fork-free families
AU - Yiyan Zhan
AU - Mei Lu
AU - Xiamiao Zhao
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.30336
UR - https://arxiv.org/abs/2609.30336
AB - A family of subsets of [n] is r-fork-free if none of its members is strictly contained in r other distinct members. For each fixed integer r≥2, we prove that the maximum size of such a family is binomn⌊ n/2⌋ (1+(2(r−1))/n+o(n^−1)). This determines the second-order term and matches the upper bound of De Bonis and Katona. Our lower bound comes from a construction on two adjacent middle levels, using a finite ordered collection of disjoint coordinate blocks and a condition on sums modulo n. For each prescribed error, the blocks are fixed before n tends to infinity, so the construction works for every sufficiently large n.
ER -