The finite Lubell threshold conjecture for posets

Let $P$ be a finite poset and let $\mathcal{F}\subseteq 2^{[n]}$. Define the Lubell function by

$$\ell(\mathcal{F})=\sum_{A\in\mathcal{F}}\frac{1}{\binom{n}{|A|}}.$$

The Lubell threshold of $P$ is

$$\lambda^{*}(P)=\limsup_{n\to\infty}\{\ell(\mathcal{F}):\mathcal{F}\subseteq 2^{[n]}\text{ and }\mathcal{F}\text{ is }P\text{-free}\}.$$

Finite Lubell threshold conjecture. Every poset has a finite Lubell threshold, that is, $\lambda^{*}(P)<\infty$ for every finite poset $P$.

The paper notes that finiteness of the Lubell threshold implies finiteness of the Turán threshold and proves finite Lubell-threshold results for certain classes of posets. The conjecture remains open for general posets.