Set families with forbidden subposets

Let $F$ be a family of subsets of $\{1,\ldots,n\}$. We say that $F$ is $P$-free if the inclusion order on $F$ does not contain $P$ as an induced subposet. The \emph{Turán function} of $P$, denoted $π^*(n,P)$, is the maximum size of a $P$-free family of subsets of $\{1,\ldots,n\}$. We show that $π^*(n,P) \le (4r + O(\sqrt{r}))\binom{n}{n/2}$ if $P$ is an $r$-element poset of height at most $2$. We also show that $π^*(n,S_r) = (r+O(\sqrt{r}))\binom{n}{n/2}$ where $S_r$ is the standard example on $2r$ elements, and that $π^*(n,B_2) \le (2.583+o(1))\binom{n}{n/2}$, where $B_2$ is the $2$-dimensional Boolean lattice.

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