12 problems

Let $P$subseteq be a finite poset. Let $\operatorname{Mon}(\mathbb{Z})$ be the set of functions $f\colon\mathbb{Z}\to\{0,1\}$ such that $f(n)=1$ and $f(-n)=0$ for all sufficiently…

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 $J_s$ 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 $P$ be a finite poset. Write … and … Define $\pi(P)=\lim_{n\to\infty}La(n,P)/\binom{n}{\lfloor n/2\rfloor}$ and…

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…

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

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