25 problems

Kirillov's conjecture. For every $u,v,w\in S_n$ and every integer $N\geq 1$,

Let $M_3$ be the convex geometric matching denoted by $M_3$ in the source, and let $\operatorname{sat}_\circlearrowright(n,M_3)$ denote its saturation number on $n$ vertices. Quadr…

Let $S_3$ be the two-edge convex geometric $3$-uniform hypergraph denoted by $S_3$ in the source, and let $\operatorname{sat}_\circlearrowright(n,S_3)$ denote its saturation number…

Let $n$ be a positive integer, let $clambda$, $cmu$, and $cnu$ be partitions of length at most $n$, and let $T_n$ consist of the triples for which $V_\nu$ occurs in…

Saturation conjecture for semisimple Lie groups. If $\lambda+\mu+\nu$ annihilates all elements of $T$ with semisimple centralizer, then

Let $\widehat{G}$ be a connected semisimple complex group, let $\widehat{T}$ be a maximal torus of $\widehat{G}$, and let $\mu_1,\mu_2,\lambda$ be weights of $\widehat{T}$. Suppose…

Full-degree vertex necessity conjecture. If

For odd $t \ge 17$, let $G_t$ be the circulant graph on $5t-10$ vertices whose distance set is … A graph is doubly saturated $R(3,t)$-good if it is $R(3,t)$-good and satisfies the…

Let $s,t \ge 3$ with $s \le t$ and $(s,t) \notin \{(3,4),(3,6)\}$. A graph is doubly saturated $R(s,t)$-good if it is $R(s,t)$-good and remains so under the relevant double-saturat…

Let $C_k$ denote the chain poset with $k$ elements, and let $\bigoplus_{j=1}^m C_{i_j}$ be the disjoint union of chains with $i_1\ge i_2\ge\dots\ge i_m$. For a poset $P$, write…

Kirillov–Klyachko conjecture. The reduced Kronecker coefficients satisfy

Let $P$ be a poset on $[p]$, and let $\dot P$ be the poset on $[p+1]$ obtained by adjoining a new element that dominates every element of $P$. Let $\text{sat}^*(n,P)$ denote the mi…

Saturation conjecture. For every $w\in W$ and every dominant $\lambda$, $\operatorname{char}(V_\lambda^w)$ is saturated.

Let $r\ge 3$ and let $H_n^{(r)}$ be the convex geometric hypergraph defined in the source. Write $\operatorname{sat}_\circlearrowright(n,M_1^{(r)})$ for the minimum number of edges…

NL-saturation conjecture. The stated equivalence holds for every $(\lambda,\mu,\nu)\in({\operatorname{Par}}_n)^3$. This is the proposed analogue of the Littlewood–Richardson satura…

Let $\mathcal{A}(\Sigma)$ be a cluster algebra with principal coefficients at the seed $\Sigma$. A Newton polytope is the convex hull of the exponent vectors of the monomials occur…

Stronger Newell-Littlewood saturation conjecture. If $N_{k\mu,k\nu,k\lambda}>0$, then there exist partitions $\alpha,\beta,\gamma$ such that

Let $[n]=\{1,2,\ldots,n\}$, let $K_{2,k}$ denote the corresponding poset, and let $\operatorname{sat}^*(n,K_{2,k})$ be the smallest size of a $K_{2,k}$-saturated family of subsets…

Let $\mathcal{C}_{2k+1}$ denote the family consisting of the cycle $C_{2k+1}$, and let $\mathrm{sat}(\mathcal{C}_{2k+1};n)$ be its saturation number on $n$ vertices. Odd-cycle satu…

Let $\alpha(k,r)$ be the parameter defined as the minimum number of edges between a specified independent transversal and its complement over all $K_r$-partite-saturated $k$-partit…

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…

Atomic saturation conjecture.

Reduced-power maximality conjecture. $(T,\Delta)$ is $\trianglelefteq_{{\rm rp}}$-maximal if and only if $(T,\Delta)$ has the ${\rm SOP}_3$.

Let $T$ be a first-order theory, let $M$ be a model of $T$, and let $\kappa$ be an infinite cardinal. A sequence $\langle a_\alpha:\alpha<\delta\rangle$ in $M$ is indiscernible whe…

Let $T_3^{\le 2}$ denote the specified binary matrix family, and let $\operatorname{sat}(n,F)$ be the minimum number of columns in an $n$-row $F$-saturated matrix. Set…