Open-problem statements, with their sources attached.

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 $K_{2n+1}$.

The Beck–Fiala conjecture*

There exists a universal constant $C > 0$ such that every set system $S_1, \dots, S_m \subseteq [n]$ of degree at most $t$ admits a colouring $\chi \colon [n] \to \{-1, +1\}$ with $\left|\sum_{j \in S_i} \chi(j)\right| \le C \sqrt{t}$ for every $i$.

Does there exist an undirected graph with 99 vertices, in which each two adjacent vertices have exactly one common neighbor, and in which each two non-adjacent vertices have exactly two common neighbors? Equivalently, every edge should be part of a unique triangle and every non-adjacent pair should be one of the two diagonals of a unique 4-cycle. The first condition is equivalent to being locally linear.

No closed-form expression that allows efficient computation of Dedekind numbers is currently known.

In particular, the Dedekind number for n = 10 is currently unknown.

Babai–Seress Conjecture (Conjecture 1.5)**: There exists an absolute constant $C$ such that the diameter of the alternating group $A_n$ satisfies $\operatorname{diam}(A_n) \leq n^C.$ Reference: L. Babai and Á. Seress, On the diameter of permutation groups, European Journal of Combinatorics 13 (1992), Conjecture 1.5

Babai–Seress Conjecture (Conjecture 1.7)**: There exists an absolute constant $C$ such that every finite simple non-abelian group $G$ satisfies $\operatorname{diam}(G) \leq (\log |G|)^C.$ Reference: L. Babai and Á. Seress, On the diameter of permutation groups, European Journal of Combinatorics 13 (1992), Conjecture 1.7