18 problems

For a poset $P$, let $La(n,P)$ and $La^*(n,P)$ denote the largest sizes of, respectively, weakly $P$-free and induced $P$-free families in the Boolean lattice $B_n$. Let $e(P)$ and…

For a finite poset $(P,\leqslant)$, let $e(P)$ be the largest integer such that the union of any $e(P)$ consecutive layers of $2^{[n]}$ is weak $P$-free, and let…

Let $d\geq3$, let $k_1,k_2,\dots,k_d$ be positive integers, let $\bigvee_2$ be the three-element fork poset, and let $w_{k,d}$ be the width of $[k]^d$. The higher-dimensional fork-…

Let $D_2$ denote the diamond poset, and for $d\geq2$ let $\pi_{d,D_2}$ and $\pi^*_{d,D_2}$ be its weak and strong density parameters in the $d$-dimensional grid. The diamond-poset…

Let $P$ be a poset of dimension at most $d$, and let $\pi_{d,P}$ and $\pi^*_{d,P}$ denote the weak and strong grid density parameters. The grid-density monotonicity conjecture. … T…

Let $P$ be a poset, let $d$ be a positive integer, and let $La([k]^d,P)$ and $La^*([k]^d,P)$ denote the weak and strong extremal numbers in the $d$-dimensional grid, normalized by…

Let $J_s$ be the permutation-pattern matrix obtained from the identity matrix by moving its last column to the beginning, and let $\bigvee_s$ be the poset with elements…

Let $\wedge_r$ be the poset on elements $a,b_1,\ldots,b_r$ with $a<b_i$ for every $i$, and let $\vee_r$ be its dual. For a family of posets $\mathcal P$ and a poset $Q$, let…

Let $Y_k$ be the poset with elements $x_1,x_2,\dots,x_k,y,z$ and relations $x_1<x_2<\dots<x_k<y,z$, and let $Y_k'$ be the poset obtained by reversing all relations. For a family of…

Forbidden fork-pair conjecture. For all $k \ge 2$ and $r \ge 2$, there is an $n_0=n_0(k,r)$ such that if $n\ge n_0$, then

Let $B$ be the butterfly poset on four elements $a,b,c,d$, with $a,b<_B c,d$. For a family $\mathcal F\subseteq 2^{[n]}$, let $Tr_{n-1}(n,B)$ denote the maximum size of a family wh…

Let $P$ be a poset, let $e(P)$ be the largest integer $k$ such that, for every $j$ and $n$, the family … is $P$-free, and let $Tr_{n-k}(n,P)$ denote the maximum size of a family wh…

Let $l(P)$ denote the height of a poset $P$, namely the length of its longest chain. For posets $P,Q$ with $l(P)>l(Q)$, let $c(Q,\mathcal F_n)$ be the number of copies of $Q$ in a…

Let $P$ and $Q$ be posets, and let $La(n,P,Q)$ denote the maximum number of copies of $Q$ in a $P$-free family ${\mathcal F}\subseteq 2^{[n]}$. A family is a full-level union if it…

Let $P$ be a fixed poset, and let $ex^*(n,P)$ denote the maximum size of a family $\mathcal{F}\subseteq\mathcal{P}[n]$ that does not strongly contain $P$, equivalently does not con…

Katona–Lu–Milans conjecture. For every poset $P$,

General poset extremal-number conjecture. The quotient

Diamond-free Lubell-function conjecture. Every such family satisfies