Crown-free families and forbidden subposets with $e(P)\in \{1,2\}$
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 $e(P)<\pi^-(P):=\liminf_{n\to\infty} \frac{La(n,P)}{\binom{n}{\lfloor \frac{n}{2}\rfloor}}$ 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 $(1.22+o(1))\binom{n}{\lfloor \frac{n}{2}\rfloor}$ 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 $La(n,P)\le(2-\delta_P+o(1))\binom{n}{\lfloor n/2\rfloor}$. 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 $K$.