18 problems

Let $\mathcal{P}$ denote the class of paths, let $\mathcal{T}_n$ denote the class of rooted trees of height at most $n$, and let $\sqsupseteq_\mathrm{FO}$ denote the first-order tr…

Saks-West conjecture. In every product $P\times Q$ of two posets, the size of a largest semiantichain equals the size of a smallest unichain covering.

Milner–Sauer conjecture. If $\operatorname{cf}(P,\le)=\lambda>\operatorname{cf}(\lambda)=\kappa$, then $P$ contains an antichain of size $\kappa$.

Unique-largest-antichain conjecture. The largest layer of $B_r[p,q]$ is its unique largest antichain.

Let $P$ be an ordered set with no infinite antichains, and let $k$ be a positive integer. Aharoni and Korman's conjecture asserts that there are $k$ chains $C_1,\ldots,C_k$ and a p…

The antichain trace conjecture. Let $k$ be a non-negative integer and let $n\geq 2k$. If $\mathcal{F}\subset 2^{[n]}$ is an antichain satisfying…

Djanković–Ivan bounded-defect conjecture.

Ferrara–Kay–Kramer–Martin–Reiniger–Smith–Sullivan conjecture. For $k\geq3$,

Let $V$ be an $n$-dimensional vector space over the finite field with $q$ elements, and let $\mathcal{L}(V)$ denote its lattice of subspaces. A family…

Kiselev–Kupavskii–Patkós' conjecture. If $1\leq 2\ell+1<n$ and $\mathcal{F}\subseteq\mathscr{P}([n])$ is a $(2\ell+1)$-union antichain, then

Let $[0,1]^n$ be equipped with the coordinatewise partial order, and let an antichain be a subset $A\subset[0,1]^n$ containing no two distinct comparable points. Write…

Let $\mathcal M$ be a class of matroids. An antichain is a collection of matroids in which no member is a minor of another. The class $\mathcal M$ is weakly fractal when the sequen…

Let $A_k$ be an antichain of size $k$, and let $f(n,c,A_k)$ be the maximum size of a color class in a $c$-coloring of $B_n=2^{[n]}$ avoiding a rainbow copy of $A_k$. Sharp exponent…

Skewed projection inequality. If $A$ is a weak antichain, then

Let $\mathcal{B}_n$ be the Boolean lattice of subsets of $[n]$. For a family $\mathcal{F}\subseteq\mathcal{B}_n$, define its convex closure by … A family is convex when…

Let $D=(N,A_0\cup A_1)$ be an arc-coloured digraph. Suppose every triangle has the form … or … and every node has exactly $d$ incoming arcs in each of $A_0$ and $A_1$. The in-degre…

A maximal $(2,3)$-antichain on $n$ points is a family of subsets in which every member has size $2$ or $3$, no member contains another, and which is maximal under these conditions.…

For a positive integer $k$, let $\mathbf{k}+\mathbf{k}$ be the poset consisting of two disjoint $k$-element chains with no comparabilities between distinct chains. Let…