58 problems

Let $[n]$ be the ground set, and let a $k$-chain be a sequence of $k$ distinct subsets of $[n]$ ordered by strict inclusion. For each family size $a$, consider the centralized fami…

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 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 $P$ be a finite poset, and let $\mathrm{sat}^{\star}([2]^n,P)$ denote its induced saturation function in the Boolean lattice. Keszegh–Lemons–Martin–Pálvölgyi–Patkós' dichotomy…

Lonc's truncated partition conjecture. If $n$ is sufficiently large and $|P|$ divides $2^n-2$, then $2^{[n]}-\{\emptyset,[n]\}$ has a $P$-partition.

Let $t\ge4$ be fixed. For each integer $r\ge2$, let $\mathcal D_{r,t}$ be the $(r,t)$-daisy, obtained by adjoining a fixed set of $r-2$ new vertices to every edge of the complete g…

Ilinca–Kahn's conjecture. The precise asymptotics satisfy

Diamond conjecture.

Quadratic-threshold conjecture. There exists a constant $c$ such that, whenever

Asymptotic sharpness conjecture. For every fixed $r\ge 1$,

Moss–Pedersen conjecture. For all sufficiently large $n$, there exists a family $\mathcal F\subseteq\binom{[2n]}{n}$ such that

Let ${\mathcal P}$ be a set of posets. Write $\mathop{}\!\mathrm{La}(n,{\mathcal P})$ and $\mathop{}\!\mathrm{La^*}(n,{\mathcal P})$ for the largest size of a family of subsets of…

Higher-order sparse Sperner conjecture. Given $k\in\mathbb N$, there exists a function $\beta^*(n)=\tilde\Theta(1/n^k)$ such that if $\beta\geq\beta^*(n)$, then almost all antichai…

Sparse Sperner conjecture. There exists a function $\beta^*(n)=\tilde\Theta(1/n)$ such that if $\beta\geq\beta^*(n)$, then almost all antichains of size $\beta N$ in $B_n$ are cont…

Erdős–Katona conjecture. The family $\mathcal{F}$ contains at least

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

Let $[n]=\{1,\ldots,n\}$, let $2^{[n]}$ be its Boolean lattice, and let $\mathcal{Q}_{2}$ denote the diamond poset. Write $ex(n,\mathcal{Q}_{2})$ for the maximum size of a family o…

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 $X_n$ be an $n$-element set, and let $2^{X_n}$ denote its power set. A fiber is a collection of sets in $2^{X_n}$ having the same value under the relevant question-answer map.…

For $k\geq 1$, let $\mathcal{B}_k(n)$ be the set of functions $f:2^{[n]}\to\mathbb{N}$ with $f(\emptyset)=0$ and $0\leq f(A\cup\{x\})-f(A)\leq k$ for every $A\subseteq[n]$ and…

Let $Q_n$ be the Boolean lattice graph, let $\mathcal{L}_k$ be the family of $k$-element subsets of $[n]$, and let $\mathcal{B}(n,k)$ be the subgraph of $Q_n$ induced by…

Adjoining-a-maximum conjecture. The quantity ${\rm sat}^*(n,P)$ is bounded as a function of $n$ if and only if ${\rm sat}^*(n,\dot P)$ is bounded as a function of $n$.

Induced saturation dichotomy conjecture. Either there exists a constant $K_P$ such that

Chain-is-best conjecture. For any $k$-element poset $P$, we have

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