35 problems

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…

Füredi's conjecture. For every positive integer $n$, the Boolean lattice $2^{[n]}$ can be partitioned into $\binom{n}{\lfloor n/2\rfloor}$ chains such that the size of each chain i…

Lonc's conjecture. If $n$ is sufficiently large and $|P|$ divides $2^n-2$, then

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 $X$ be an $n$-element set, and let $\binom{X}{\bullet}$ denote the Boolean lattice of subsets of $X$. For integers $t,k\geq 1$ with $n=tk$, partition $X$ into pairwise disjoint…

Ilinca–Kahn's conjecture. The precise asymptotics satisfy

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…

Let $P$ be a finite poset and let $\mathcal{B}_n=2^{[n]}$ be the Boolean lattice. Write $La(n,P)$ for the largest size of a $P$-free subfamily of $\mathcal{B}_n$. Define $e(P)$ to…

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

Let $[n]=\{1,\ldots,n\}$, let $2^{[n]}$ denote its power set, and let $\mathcal{F}$ be a maximal intersecting subfamily of $2^{[n]}$. For each $A\subseteq[n]$, let $e_A$ be the cor…

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

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…

Let $A_k$ be an antichain of size $k$, and let $\bigvee_s$ and $\bigwedge_s$ denote the posets consisting respectively of a common minimal or maximal element together with $s$ inco…

Quadratic near-perfect packing conjecture. There exists a constant $c$ such that, if

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

Let $t_{\mathrm{end}}$ denote the time at which the diffusion limited aggregation process reaches its terminal stage in the Boolean lattice on $n$ elements. Subexponential end time…

Let $k$ be a positive integer, and let $n_0$ be an integer. Equip the product poset $\{0,1,\ldots,k\}^n$ with the coordinatewise order. A family is centered when its elements are c…

Let $n$ be a positive integer, let $\mathcal{B}_n$ be the Boolean lattice, and let $\mathcal{F}\subseteq\mathcal{B}_n$ be an induced-$\mathcal{D}_2$-saturated family that avoids bo…

Gruslys–Leader–Tomon conjecture. The Boolean lattice $2^{[n]}$ can be partitioned into copies of $P$ and a remainder of at most $c=c(P)$ elements.

Almost-tiling conjecture. There exists a constant $c=c(P)$ such that for every positive integer $n$, there is a set $S\subseteq 2^{[n]}$ with $|S|\leq c$ such that…

For positive integers $n$ and $k$, let $\mathcal{B}_{n,k}$ be the graph with vertex set $\binom{[n]}{k}\cup\binom{[n]}{k+1}$ and edges given by inclusion, and let…

Uniform cutset conjecture.