48 problems

Kirillov–Klyachko's conjecture. The reduced Kronecker coefficients satisfy saturation:

Let $P$ and $Q$ be patterns, and let $P\otimes Q$ denote their Kronecker product. Write $\operatorname{sat}(n,P)$ for the square saturation function and $\operatorname{ssat}(n,Q)$…

A permutation matrix is a square $0$-$1$ matrix with exactly one $1$ in each row and column. The saturation function $\operatorname{sat}(n,P)$ is the minimum number of $1$ entries…

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…

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 $n$ 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…

Let $G$ be a reductive group, let $\widehat{G}$ be its Langlands dual group, and let $\mu_1,\ldots,\mu_r$ be dominant weights of $\widehat{G}$. Suppose each $\mu_i$ is a sum of dom…

Full-degree vertex necessity conjecture. If

The theory $ACFA_p$ denotes the theory of existentially closed difference fields of characteristic $p>0$. The $\b5$-existence property over the fixed field asks whether every $\b5$…

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…

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

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 $i_1\ge i_2\ge\dots\ge i_m$. Assume that $i_1>i_m$, so…

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…

Let $T$ be any $k\times k$ triangular $0$-$1$ matrix with all ones on the diagonal. For an $m\times n$ matrix, let $\operatorname{sat}(m,n,T)$ be the minimum number of $1$ entries…

Let … For an $m\times n$ matrix, let $\operatorname{sat}(m,n,P)$ be the minimum number of $1$ entries in a $P$-avoiding matrix that becomes non-avoiding after changing any $0$ to a…

Let $P$ be any pattern, and let $P'$ be derived from $P$ by adding empty rows and columns in the interior, meaning strictly between the first and last row and column. Let…

Erdős–Sós conjecture. The maximum number of hyperedges in a $k$-uniform hypergraph without a bow-tie $B_k$ is

Let $w$ be a permutation, and let $\underline\lambda$ and $\underline\mu$ be partition-tuples. For a positive integer $N$, write $N\underline\lambda=(N\lambda_1,N\lambda_2,\dots)$…

Continuous-model-theory analogue conjecture. Results analogous to the main theorems of the paper should hold in continuous model theory.

Optimality conjecture. The assumptions on the ideal $$ in this theorem are far from optimal.

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…

Let $[n]=\{1,\ldots,n\}$, let $2^{[n]}$ be its subset lattice, and let $\text{sat}^*(n,P)$ denote the minimum size of an induced $P$-saturated family of subsets of $[n]$. Let…