Typical intersecting families at n=2k+1 and n=2k+2
Abstract
A family of sets is intersecting if every two members intersect, and trivial if all its members contain a common element. We determine the typical structure of k-uniform intersecting families on 2k+1 and 2k+2 elements as k→∞. For n=2k+2, we prove that almost all intersecting families are trivial and that their number is (2k+2+o(1)) 2binom2k+1k−1. Together with Yang's recent result for n≥ 2k+3, this settles a conjecture of Balogh, Garcia, Li, and Wagner. For n=2k+1, almost all intersecting families are nontrivial. We prove that, as conjectured by the same authors, a typical intersecting family is close to a full star: its members outside the star form components of size at most two in the graph joining sets that intersect in k−1 elements. We also obtain an asymptotic formula for the number of intersecting families in this case, with an explicit second-order term in the exponent. Our proof combines Sapozhenko's graph container method and stability in Kneser graphs to control families far from every star, and a polymer model and cluster expansion to enumerate families close to a fixed star.
Related
BibTeX
@misc{li2026typical,
title = {{Typical intersecting families at $n=2k+1$ and $n=2k+2$}},
author = {Lina Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02119},
eprint = {2610.02119},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02119},
}
RIS
TY - UNPB
TI - Typical intersecting families at n=2k+1 and n=2k+2
AU - Lina Li
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.02119
UR - https://arxiv.org/abs/2610.02119
AB - A family of sets is intersecting if every two members intersect, and trivial if all its members contain a common element. We determine the typical structure of k-uniform intersecting families on 2k+1 and 2k+2 elements as k→∞. For n=2k+2, we prove that almost all intersecting families are trivial and that their number is (2k+2+o(1)) 2^binom2k+1k−1. Together with Yang's recent result for n≥ 2k+3, this settles a conjecture of Balogh, Garcia, Li, and Wagner. For n=2k+1, almost all intersecting families are nontrivial. We prove that, as conjectured by the same authors, a typical intersecting family is close to a full star: its members outside the star form components of size at most two in the graph joining sets that intersect in k−1 elements. We also obtain an asymptotic formula for the number of intersecting families in this case, with an explicit second-order term in the exponent. Our proof combines Sapozhenko's graph container method and stability in Kneser graphs to control families far from every star, and a polymer model and cluster expansion to enumerate families close to a fixed star.
ER -