AI 中文总结
本文研究偏序集参数$e(P)$与$\pi^-(P)$的关系，新增$e(P)=1$的偏序集，构造弱$O_6$-自由族，证明相关不等式并构造$e(P)=2$且下密度任意大的偏序集。
AI 中文摘要
弱$P$-自由族$\mathcal{F}\subseteq 2^{[n]}$的最大规模记为$La(n,P)$，令$e(P)$为使得$2^{[n]}$任意$k$个连续层的并均为弱$P$-自由族的最大整数$k$。近年已发现多个满足$e(P)<\pi^-(P):=\liminf_{n\to\infty} \frac{La(n,P)}{\binom{n}{\lfloor \frac{n}{2}\rfloor}}$的偏序集。本文新增多个$e(P)=1$的偏序集至该列表；定义规模至少为$(1.22+o(1))\binom{n}{\lfloor \frac{n}{2}\rfloor}$的弱$O_6$-自由族，其中$O_6$为六元冠偏序集；还证明存在无穷多满足$1=e(P)<\pi^-(P)$且对该性质极小的偏序集$P$。最后探究$e(P)$与$\pi^-(P)$的差距，证明对每个固定有限偏序集$P$且$e(P)=1$，存在常数$\delta_P>0$使得$La(n,P)\le(2-\delta_P+o(1))\binom{n}{\lfloor n/2\rfloor}$，值2是最优的：具$e(P)=1$的显式顶点-边关联偏序集的$\pi^-(P)$值趋于2；反之，对任意$K>0$，本文构造了一个有限偏序集$P$满足$e(P)=2$且下密度大于$K$。
英文摘要
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 $K$.
