An upper bound on the size of diamond-free families of sets

Abstract: Let $La(n,P)$ be the maximum size of a family of subsets of $[n]={1,2,...,n}$ not containing $P$ as a (weak) subposet. The diamond poset, denoted $B_{2}$, is defined on four elements $x,y,z,w$ with the relations $x&lt;y,z$ and $y,z&lt;w$. $La(n,P)$ has been studied for many posets; one of the major open problems is determining $La(n,B_{2})$. Studying the average number of sets from a family of subsets of $[n]$ on a maximal chain in the Boolean lattice $2<sup>{[n]}$ has been a fruitful method. We use a partitioning of the maximal chains and introduce an induction method to show that $$La(n,B_{2})\leq(2.20711+o(1))\binom{n}{\left\lfloor \frac{n}{2}\right\rfloor }$$, improving on the earlier bound of $$(2.25+o(1))\binom{n}{\left\lfloor \frac{n}{2}\right\rfloor }$$ by Kramer, Martin and Young.