Supersaturation in union-closed families of sets

Christopher Bouchard

Supersaturation in union-closed families of sets

Combinatorics 2026-07-29 v1

Authors: Christopher Bouchard

Abstract

Let $k$ and $n$ be positive integers such that $2 \leq k \leq n+1$ . We prove that the number of $k$ -chains in a union-closed family with universe $[n]$ and size $m$ is minimized when its member sets are largest possible. We also show that, whenever the minimum is nonzero and $m>n$ , there are no other minimizing families.

Cite

@article{arxiv.2607.26425,
  title  = {Supersaturation in union-closed families of sets},
  author = {Christopher Bouchard},
  journal= {arXiv preprint arXiv:2607.26425},
  year   = {2026}
}