1,922 problems

Problem. Is every finite uniformly dense matroid of rank $3$ cyclically orderable?

For a permutation $\sigma$ of $\{1,2,\dots,n\}$ and a permutation $\beta$ of $\{1,2,\dots,k\}$, say that $\sigma$ contains $\beta$ as a pattern if there are indices…

For every connected graph $G$ with $n=|V(G)|$ vertices, the edges of $G$ admit a decomposition into at most $\left\lceil\frac{n}{2}\right\rceil$ paths.

Let $\mathcal{F}$ be a finite family of distinct finite sets that is union-closed, i.e. … and suppose $\mathcal{F}\neq\varnothing$ and $\mathcal{F}\neq\{\varnothing\}$. For an elem…

Let $\pow\geq2$, $k\geq1$, and let $a_1,\ldots,a_k,b$ be positive integers. Write $[r]_q:=1+q+\dots+q^{r-1}$ and, for an integer $\pow$, write…

Let $n=4k+7$ with $k\geq 2$, and let $\operatorname{Cone}=\operatorname{Cone}(\operatorname{Cone}_{2k+1}\cup\operatorname{Cone}_{2k+3})$. Cone asymptotic-size conjecture. Is … Nume…

Let $n=4k+7$ and let $\mathcal{D}$ be the constructed maximal commutative algebraic system in $\mathcal{P}_n(*)$. $\mathcal{D}$ asymptotic-size conjecture. Is … Numerical checks th…

Let $G_n$ be the graph associated with $S(0,n)$, and let $K_{\ell,v,n}$ denote the number of paths of length $\ell$ in $G_n$ that start at a vertex labeled $v$ and end at a vertex…

Let $(s_n)$ be an appropriate sequence of positive integers, and let $G_n$ be the action graph associated with $s_n$. For $n\geq 1$, write $z_n$ for the number of new vertices labe…

Let $f_{CS}(x)$ be the generating function for the cyclic shift strategy of length $n$, and let $f_S(x)$ be the generating function for a deranged strategy $S$, meaning a strategy…

Unified row-shifted second-power lattice conjecture. For $n \geq 2$,

For each integer $m\geq 4$, let $f_m$ and $h_m$ be the polynomials defined in the paper, and let the ULC inequality refer to the ultra-log-concavity inequality for their coefficien…

Negative-coefficient conjecture. In the remaining 33 pairs of the polynomials $A_{abcd}$ and $B_{2-a,2-b,2-c,2-d}$, including the case where $p\neq 2$ for $a=b=c=d=1$, one of the t…

Sub-leading Laurent coefficient conjecture. For every $d\ge5$ and every $m\ge2$,

Let $C$ be a set, let $\boldsymbol{a}$ be a transitive $n\times n$ matrix with entries in $C$, and let $\mathbf r=\{r^{(c)}\}_{c\in C}\subset \mathfrak{g}\otimes\mathfrak{g}$ be a…

Let $xi_{\ell,r,c}$ denote the integer in the reduced $Gr_2$-simplex in layer $\ell$, row $r$, and NE-SW diagonal $c$, and set $\xi_{\ell,r,c}=0$ when there is no entry in that pos…

Arrange the entries in each layer of the $Gr_2$-simplex as coefficients of a polynomial in $a,b,c$ according to the arrangement described in the paper. Divisibility conjecture. Eac…

Recurrence and coefficient conjecture. The linear recurrence corresponding to $k$ has order $\left\lfloor k/2\right\rfloor+3$, and each coefficient $c_j(q)$ is a linear polynomial…

For each $n\geq 2$, let $\mathcal{G}_n$ be the set of Gilbreath sequences of length $n$, let $K_S$ denote the valid-extension set associated with $S$, and let $U_n$ be the doubling…

Coprime dice relabeling conjecture. There are no ways to relabel a die of size $r$ and a die of size $s$ without changing the frequencies of their sums.

Standard-Vandermonde determinant formula. The first appearance value is

Standard-Vandermonde value conjecture. For $n \geq 2$,

Multiplication-table lattice value conjecture. For $n \geq 2$,

Harmonic APD relation. For $n \geq 1$,

Hilbert-matrix first appearance value conjecture. For $n \geq 2$,