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 $e(P)<π^-(P):=\liminf_{n\to\infty} \frac{La(n,P)}{\binom{n}{\lfloor \frac{n}{2}\rfloor}}$ 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 $1=e(P)<π^-(P)$ that are minimal with respect to this property. Finally, we consider how far apart $e(P)$ and $π^-(P)$ can be. We prove that for every fixed finite poset $P$ with $e(P)=1$, there is a constant $δ_P>0$ such that $La(n,P)\le(2-δ_P+o(1))\binom{n}{\lfloor n/2\rfloor}$. The value 2 is optimal: explicit vertex-edge incidence posets with $e(P)=1$ have $π^-(P)$ 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 $K$.

Disclosure

“AI declaration. The constructions in this manuscript were obtained through the use of Chat- GPT 5.6. The arguments have been reworked and carefully verified by the authors, who take full responsibility for the content of the manuscript. References [1] M. Axenovich, R.R. Martin, B. Patkós, Extremal Poset Theory, In: A. Gagarin, R. Beh”

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Classification

Substantial mathematical content or result generation

Multiplier

10

Verified

2026-08-23T05:22:34.825367Z

Structural counts