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 $P$ 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…

Let $P$ be a poset, and consider the induced saturation function $\mathrm{sat}^{\star}([t]^n,P)$ whenever it is defined. Monotonicity conjecture. The function…

Let $P$ be a poset and fix $t\in\mathbb{Z}_{\geq 2}$. Write $\mathrm{sat}^{\star}([t]^n,P)$ for the induced saturation function of $P$ in the $t$-ary hypergrid. Linear-growth dicho…

Let $P$ be a finite poset, and let $w^{\star}(P)$ denote its cube width. Bastide–Groenland–Ivan–Johnston's cube-width conjecture. One has … The supplied text states that this conje…

Hypercube saturation conjecture. For every $k\geq 2$,

Structural conjecture for minimum diamond-saturated families. If

Bounded-or-linear induced saturation conjecture. For every finite poset $\mathcal P$, either

The linear-or-bounded conjecture. For every finite poset $\mathcal P$, the induced saturated number $\operatorname{sat}^*(n,\mathcal P)$ is either bounded, or at least linear in…

A poset is a set equipped with a transitive, antisymmetric, and reflexive relation. For a poset $\mathcal{P}$, let $sat^*(n,\mathcal{P})$ denote the minimum size of an induced-…

Let $\mathcal P_1$ and $\mathcal P_2$ be finite posets with bounded saturation number, and let $\mathcal P_2*\mathcal P_1$ denote their gluing, obtained by placing every element of…

Let $\mathcal P_1$ and $\mathcal P_2$ be finite posets, and let $\mathcal P_2*\mathcal P_1$ denote their gluing, obtained by placing every element of $\mathcal P_1$ below every ele…

Let $K_{n_1,\dots,n_k}$ denote the complete poset with consecutive layers of sizes $n_1,\dots,n_k$, where $n_1,\dots,n_k$ are positive integers and at least one $n_i$ is not equal…

Let $\mathcal P$ be a finite poset, and let $\dot{\mathcal P}$ be the poset obtained by adding a new element greater than every element of $\mathcal P$. Let…

Let $\mathcal P$ be a finite poset, and let $\operatorname{sat}^*(n,\mathcal P)$ denote the minimum size of a family of subsets of $[n]$ that is induced $\mathcal P$-saturated. Lin…

Let $s\geq t\geq 2$ be fixed, and let $\mathcal{K}_{s,t}$ be the complete bipartite poset with $s$ upper-layer and $t$ lower-layer pairwise incomparable vertices, every upper-layer…

For $s\geq 2$, let $\mathcal{K}_{s,t}$ be the complete bipartite poset with $s$ upper-layer and $t$ lower-layer pairwise incomparable vertices, every upper-layer vertex larger than…

For $n\in\mathbb{N}$, let $\mathcal{K}_{2,2}$ be the complete bipartite poset with two incomparable elements in each layer, every upper-layer element larger than every lower-layer…

Keszeg̋h's lower-bound conjecture. For the posets under consideration,

Djanković–Ivan bounded-defect conjecture.

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