Three layer $Q_2$-free families in the Boolean lattice

Abstract

We prove that the largest $Q_2$-free family of subsets of $[n]$ which contains sets of at most three different sizes has at most $(3+2\sqrt{3})N/3+o(N)\approx 2.1547N+o(N)$ members, where $N=\left(\genfrac{}{}{0ex}{}{n}{\lfloor n/2\rfloor }\right)$. This improves an earlier bound of $2.207N+o(N)$ by Axenovich, Manske, and Martin.

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