Open-problem statements, with their sources attached.

For $N = 6$ and $D = 5$, does there exist no solution to the monochromatic quantum graph equation system over $\mathbb{Z}$?

For $N = 6$ and all $D \geq 3$, does there exist no solution to the monochromatic quantum graph equation system over $\mathbb{Z}$?

For $N = 8$ and $D = 3$, does there exist no solution to the monochromatic quantum graph equation system over $\mathbb{Z}$?

For $N = 10$ and $D = 3$, does there exist no solution to the monochromatic quantum graph equation system over $\mathbb{Z}$?

For all even $N \geq 6$ and $D \geq 3$, does there exist no solution to the monochromatic quantum graph equation system over $\mathbb{Z}$?

For $N = 6$ and $D = 3$, does there exist no solution to the monochromatic quantum graph equation system over $\mathbb{Z}$ with weights in $\{-1, 0, 1\}$?

For $N = 6$ and $D = 5$, does there exist no solution to the monochromatic quantum graph equation system over $\mathbb{Z}$ with weights in $\{-1, 0, 1\}$?

For $N = 6$ and all $D \geq 3$, does there exist no solution to the monochromatic quantum graph equation system over $\mathbb{Z}$ with weights in $\{-1, 0, 1\}$?

For $N = 8$ and $D = 3$, does there exist no solution to the monochromatic quantum graph equation system over $\mathbb{Z}$ with weights in $\{-1, 0, 1\}$?

For $N = 10$ and $D = 3$, does there exist no solution to the monochromatic quantum graph equation system over $\mathbb{Z}$ with weights in $\{-1, 0, 1\}$?

For all even $N \geq 6$ and $D \geq 3$, does there exist no solution to the monochromatic quantum graph equation system over $\mathbb{Z}$ with weights in $\{-1, 0, 1\}$?

For a graph $G$, we define $\Delta(G)$ to be the maximum degree, $\omega(G)$ to be the size of the largest clique subgraph, and $\chi(G)$ to be the chromatic number. Reed's omega, delta, and chi conjecture states that $\chi(G) \leq \lceil \frac{1}{2}(\omega(G) + \Delta(G) + 1) \rceil.$

For a finite graph $G$, we define $\Delta(G)$ to be the maximum degree, $\omega(G)$ to be the size of the largest clique subgraph, and $\chi(G)$ to be the chromatic number. Reed's omega, delta, and chi conjecture states that $\chi(G) \leq \lceil \frac{1}{2}(\omega(G) + \Delta(G) + 1) \rceil.$

The simplest open case is when $\Delta(G) = 6$ and $\omega(G) = 2$.

For any tree $T$ with $n$ edges, the complete graph $K_{2n+1}$ decomposes into $2n+1$ edge-disjoint copies of $T$.

A "copy" of $T$ is the image $T.\text{map}(f_i)$ of $T$ under a vertex embedding $f_i : V \hookrightarrow \text{Fin}(2n+1)$; the copies are pairwise edge-disjoint and together cover every edge of $K_{2n+1}$.

Let $T_N = \sum_{k=1}^{N} k = \frac{N(N+1)}{2}$. If $T_N$ is even (equivalently $N \equiv 0 \pmod 4$ or $N \equiv 3 \pmod 4$), then under optimal play the game Catch-Up($\{1, \ldots, N\}$) ends in a draw.

Every even number greater than 4208 is the sum of two twin primes.

Kurepa's conjecture

For all $n$, $!n\not\equiv 0 \mod n$

This appears as B44 "Sums of factorials." in Unsolved Problems in Number Theory by Richard K. Guy

This statement can be reduced to the prime case only.

An equivalent formulation in terms of the gcd of $n!$ and $!n$.