11 problems

Let $P$ be a finite poset. For a family $\mathcal F\subseteq 2^{[n]}$, let $La(n,P)$ be the largest size of a $P$-free family, and let $La^*(n,P)$ be the largest size of an induced…

Diamond conjecture.

Two-middle-layers conjecture. The greatest asymptotic density of a diamond-free family is achieved by taking the union of the two middle layers.

Let $P$ be a finite connected poset. Let $e(P)$ and $e^*(P)$ be the consecutive-level parameters, and let $d(P)$ and $d^*(P)$ be the minimum ratios defined over connected subposets…

Let $P$ be a finite poset. Let $e(P)$ and $e^*(P)$ be the consecutive-level parameters for, respectively, $P$-free and induced $P$-free families in $2^{[n]}$. The counting conjectu…

Let $P$ be a poset and $\varepsilon>0$. Let $La(n,P)$ and $La^*(n,P)$ be the largest sizes of, respectively, $P$-free and induced $P$-free families in $2^{[n]}$. Let $e(P)$ and…

The asymptotic extremal subspace-poset conjecture. For any integer $n$ and poset $P$, we have

The asymptotic extremal poset conjecture. For any integer $n$ and poset $P$, we have

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

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

Let $[n]=\{1,\ldots,n\}$, let $2^{[n]}$ be the Boolean lattice, and let ${\rm La}(n,P)$ denote the maximum size of a family of subsets of $[n]$ containing no extension of the poset…