23 problems

For a fixed integer $k\geq 3$, let $\mathcal{A}_k$ be the antichain with $k$ elements, and let $\mathrm{sat}^{\star}([2]^n,\mathcal{A}_k)$ denote its induced saturation function in…

Let $\mathcal{P}(n,p)$ be the random set-system obtained by including each subset of an $n$-element set independently with probability $p$, and let $k$ be the fixed chain-length pa…

Pelekis–Vlasák's conjecture. There exists a $k$-antichain $A\subseteq[0,1]^n$ satisfying

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.

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

Largest-antichain bound. For every $n\in\mathbb{Z}^+$,

Let $[0,1]^n$ be partially ordered coordinatewise: $\textbf{x}\leq\textbf{y}$ when $x_i\leq y_i$ for every $i=1,\dots,n$. An antichain is a subset $A\subseteq[0,1]^n$ containing no…

Let $n\geq 2$ and let $k$ be a positive integer. A $k$-antichain is a set $A\subset[0,1]^n$ such that every chain $C\subset[0,1]^n$ satisfies $|A\cap C|\leq k$. Write…

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 $n\geq 1$. An antichain in $[0,1]^n$ is a subset whose distinct points are incomparable in the coordinatewise order, and let $\mathcal H^{n-1}$ denote $(n-1)$-dimensional Hausd…

Let $\mathcal{P}(n)$ denote the Boolean lattice of all subsets of $[n]$, and let $\operatorname{ma}(\mathcal{P}(n))$ be the number of maximal antichains in $\mathcal{P}(n)$. Ilinca…

A graph class is finitely defined if it has finitely many minimal forbidden induced subgraphs. An infinite antichain is an infinite collection of graphs in which no graph is an ind…

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…