An Improved Lower Bound for Diamond-Free Families

Casey Tompkins

An Improved Lower Bound for Diamond-Free Families

Combinatorics 2026-07-10 v1

Authors: Casey Tompkins

Abstract

We construct a diamond-free family in the Boolean lattice whose size is asymptotically larger than the union of two middle layers. Denote the diamond poset by $Q_2$ and let $La(n,Q_2)$ be the maximum size of a family in $2^{[n]}$ containing no weak copy of $Q_2$ . We prove $La(n,Q_2) \ge (c+o(1))\binom{n}{\lfloor n/2\rfloor}$ , where $c \approx 2.147908$ . In particular, this disproves the diamond conjecture.

Cite

@article{arxiv.2607.09497,
  title  = {An Improved Lower Bound for Diamond-Free Families},
  author = {Casey Tompkins},
  journal= {arXiv preprint arXiv:2607.09497},
  year   = {2026}
}