On union-closed families with prescribed number of k-sets

@misc{jafari2026unionclosed,
  title = {{On union-closed families with prescribed number of $k$-sets}},
  author = {Amir Jafari},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.11358},
  eprint = {2609.11358},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.11358},
}

TY  - UNPB
TI  - On union-closed families with prescribed number of k-sets
AU  - Amir Jafari
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2609.11358
UR  - https://arxiv.org/abs/2609.11358
AB  - Fix positive integers N,k,n with n≥ k. We seek the minimum number of members of size at least n in a finite family of finite sets closed under union and containing exactly N distinct sets of size k. This problem is a specialization of the Leck--Roberts--Simpson weighted conjecture: assign weight one to sets of size at least n and zero to smaller sets. The predicted minimizer consists of the unions of nonempty subfamilies of the first N k-subsets of the natural numbers, ordered by their largest elements and, when these agree, by their increasing lists lexicographically. For an integer t≥ 1, call the range binomn+t−1k<N≤binomn+tk the t-th strip. We prove the layered conjecture throughout the first strip, and throughout the second strip for k=3. For arbitrary k, we prove the second strip for families of subsets of an (n+2)-element set. For k,t≥ 3, we prove the t-th strip for families of subsets of an (n+t)-element set whenever n≥(t+1)(k−1). With no restriction on the ground set, we prove it for k≥ 3 and t≥ 2 whenever n>5/2 k^2t. For sufficiently large k, we obtain a sufficient bound of order k^2t/log k, uniformly in t≥2.
ER  -