12 problems

For all finite graphs $G$ and $H$, the quantum chromatic number of their categorical product satisfies $\chi_q(G\times H)=\min\{\chi_q(G),\chi_q(H)\}$, where $G\times H$ has vertex…

For every admissible parameter triple $(n,d,s)$ and every relevant witness family $\mathcal{F}$, the uniform witness conjecture asserts that…

Determine whether the following sharp upper-density bound holds: if $A\subseteq\mathbb{N}$ satisfies $ab+1$ is not squarefree for every $a,b\in A$, then…

For every integer $n\ge 1$ and every tiling $\mathcal{T}$ of $\mathbb{R}^n$ by unit $n$-dimensional cubes, there exist distinct cubes $C,C'\in\mathcal{T}$ such that $C\cap C'$ is a…

Determine whether Karpenkov's analytic extension of the $\sin^2$-algorithm is eventually periodic for every cubic irrationality of signature $(1,1)$, that is, for every real cubic…

Let $U(n)$ be the maximum number of unordered pairs of points at Euclidean distance $1$ among any set of $n$ points in the plane, that is,…

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The sources identify Erdős problem $\#608$ as an extremal problem involving a claimed threshold of $2/9$, but do not provide the underlying objects or define the extremal quantity…

Let $R$ be an operator-only Pure Recursive Calculus with a recursor rule of the form … The step argument $s$ is unrestricted, and an internally definable measure means a measure sa…

Let $T_{1}$ be the base theory for list concatenation. For a finite sequence of pairwise distinct list variables $\vec{X}=(X_{1},\dots,X_{m})$ and a sequence of non-zero natural nu…

In three-dimensional space, consider packings of congruent non-overlapping spheres, and let their density be the proportion of space occupied by the spheres, interpreted as the asy…