25 problems

Let $F:\{0,1\}^{\mathbb{Z}^2}\to\{0,1\}^{\mathbb{Z}^2}$ be the global map of Conway's Game of Life, and let…

For each $n\in\{4,5\}$, let $S$ be a nondegenerate Euclidean $n$-simplex, and let $M(S)$ be the simplex formed by trisecting the dihedral angles of $S$. If $M(S)$ is regular, then…

Does there exist a biplanar graph $G=(V,E)$ such that $|V|=19$ and $\alpha(G)=2$? Here, biplanar means that $E$ can be partitioned into the edge sets of two planar graphs on the co…

For every prime $p\ge 3$, there exist distinct primes $p_1$ and $p_2$ such that $p=2p_1-p_2$; equivalently,…

For every pair of integers $f\ge 1$ and $g>f$, define the increasing sequence $S_{f,g}=(a_n)_{n\ge 1}$ by $a_1=f$, $a_2=g$, and, recursively, $a_{n+1}$ is the smallest integer grea…

For every knot $K$, there exists a minimal-crossing diagram $D$ of $K$ and a crossing of $D$ such that changing that crossing produces a knot $K'$ satisfying $u(K')=u(K)-1$, where…

For every integer $n\ge 4$ and every rotation system $R$ on an $n$-element set, at least…

For every nontrivial planar graph $G$, the bondage number satisfies $b(G)\leq \Delta(G)+1$, where $b(G)=\min\{\lvert F\rvert:F\subseteq E(G),\ \gamma(G-F)>\gamma(G)\}$ and…

For every integer $n\ge 1$, every monotone submodular function $f:2^{[n]}\to\mathbb{R}_{\ge 0}$, and every marginal vector $x\in[0,1]^n$ with $f^{++}(x)>0$, the correlation gap und…

Let $G_{\mathrm{frb100\text{-}40}}=(V,E)$ be the publicly specified $4{,}000$-vertex Model-RB benchmark graph. Determine its independence number…

Every planar graph $G$ contains a vertex subset $S\subseteq V(G)$ such that the induced subgraph $G[S]$ is a forest and $|S|\geq \frac{|V(G)|}{2}$.

For every integer $n\ge 1$ and every finite, nonzero Borel measure $\mu$ on $\mathbb{R}^n$ that is absolutely continuous with respect to Lebesgue measure, there exist $n$ affine hy…

Let $\Delta$ be a triangulation of a simply connected polygonal domain $\Omega\subset\mathbb{R}^2$, let $f_1^\circ$ and $f_0^\circ$ denote the numbers of interior edges and interio…

Determine whether $\lambda^*(L)\ge 0$ for every $L>0$, where $\lambda^*(L)=\inf\left\{Q(f)/\lVert f\rVert_2^2:0\ne f,\ \operatorname{supp}f\subseteq[-L,L]\right\}$ and $Q$ is Weil’…

For each integer $n\geq 1$, determine … where … In particular, determine the exact value of $M_{\mathrm{bal}}(8)$; the supplied report claims $M_{\mathrm{bal}}(8)=116$, excluding…

Consider the separable constrained optimization problem $\min_{x_1,x_2,x_3}\ f_1(x_1)+f_2(x_2)+f_3(x_3)$ subject to $A_1x_1+A_2x_2+Ix_3=b$, where the first two objective functions…

For every integer $m\ge 1$, let $X_m^4=\{x_0^m+x_1^m+x_2^m+x_3^m+x_4^m+x_5^m=0\}\subset\mathbb{P}^5$ be the Fermat fourfold. The Hodge conjecture asserts that every rational Hodge…

For each integer $k\ge 1$, determine the smallest integer $n\ge k$ such that $n(n-1)(n-2)\cdots(n-k)$ divides the central binomial coefficient $\binom{2n}{n}$; equivalently, determ…

A position in $4 \times n$ Chomp is a quadruple $(a,b,c,d)$ of non-negative integers with $a\geq b\geq c\geq d\geq 0$; a P-position is a position from which the previous player has…

Let $p$ and $q$ be coprime integers with $q$ odd. For the periodic problem $\Sigma(p/q)$, let $\gamma_k(p/q)$ denote the length of the gap with label $k$. Gap-decay and extremal-fr…

Let $m$ be an odd integer with $m\geq 5$, and let $W\in\mathbb{Z}_2*\mathbb{Z}_4$. Let $\tau_W(\lambda)$ denote the trace polynomial associated with $W$. The trace-polynomial nonex…

Let $B_3$ be the three-strand braid group, let $\Sigma_3^{\scriptstyle *}$ denote its dual generating set, and let $\mathcal{S}(B_3,\Sigma_3^{\scriptstyle *})$ and…

Let $t_{2}(2,q)$ be the smallest size of a complete arc in $\mathrm{PG}(2,q)$, and let the bounds denoted in the source by … be the explicit upper bounds obtained from the computed…

Let $a_\rho^{(3)}$ be the coefficients in the monomial expansion of $g_3$. Kerov's numerical conjecture. For $k=3$, the values of $500\cdot5!\cdot7!\,a_\rho^{(3)}$ are given by the…

Let $c_\rho^{(3)}$ be the coefficients in the monomial expansion of $f_3$. Goulden–Rattan's numerical conjecture. For $k=3$, the values of $2\cdot 6!\cdot 8!\,c_\rho^{(3)}$ are giv…