Crown-free families and forbidden subposets with $e(P)\in \{1,2\}$

Abstract

The maximum size of a weak $P$-free family $\mathcal{F}\subseteq 2^{[n]}$ is denoted by $La(n,P)$. Let $e(P)$ denote the maximum integer $k$ such that the union of any $k$ consecutive layers of $2^{[n]}$ is weak $P$-free. In recent years, multiple examples of posets with $e(P)<{\pi }^{-}(P):={\mathrm{liminf}}_{n\to \infty }\frac{La(n,P)}{\left(\genfrac{}{}{0ex}{}{n}{\lfloor \frac{n}{2}\rfloor }\right)}$ have been found. We add several further posets with $e(P)=1$ to this list. We define a family $\mathcal{F}\subseteq 2^{[n]}$ of size at least $(1.22+o(1))\left(\genfrac{}{}{0ex}{}{n}{\lfloor \frac{n}{2}\rfloor }\right)$ that is weak $O_6$-free, where $O_6$ is the six-element crown poset. We also show an infinite set of posets $P$ with $1=e(P)<{\pi }^{-}(P)$ that are minimal with respect to this property. Finally, we consider how far apart $e(P)$ and ${\pi }^{-}(P)$ can be. We prove that for every fixed finite poset $P$ with $e(P)=1$, there is a constant ${\delta }_P>0$ such that $La(n,P)\le (2-{\delta }_P+o(1))\left(\genfrac{}{}{0ex}{}{n}{\lfloor n/2\rfloor }\right)$. The value 2 is optimal: explicit vertex-edge incidence posets with $e(P)=1$ have ${\pi }^{-}(P)$ values tending to $2$. In contrast, for every $K>0$ we construct a finite poset $P$ with $e(P)=2$ and lower density greater than $K$.