An Improved Lower Bound for Diamond-Free Families

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 $La(n,Q_2)\ge (c+o(1))\left(\genfrac{}{}{0ex}{}{n}{\lfloor n/2\rfloor }\right)$, where $c\approx 2.147908$. In particular, this disproves the diamond conjecture.