Open-problem statements, with their sources attached.

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

Let $k ≥ 2$. Does $((n+k)!)^2∣(2n)!$ hold for infinitely many $n$?

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

It is open even for $k = 2$. Let $k = 2$. Does $((n+k)!)^2∣(2n)!$ hold for infinitely many n?

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 $\{-1, 0, 1\}$?

Are there infinitely many pairs of integers $n < m$ such that $\binom{2n}{n}$ and $\binom{2m}{m}$ have the same set of prime divisors?

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 $\chi(G) \leq \lceil \frac{1}{2}(\omega(G) + \Delta(G) + 1) \rceil.$

Let $\epsilon>0$. Does there exist $A\subseteq \mathbb{N}$ such that the lower density of $A+A$ is at least $1-\epsilon$ and yet $1_A\ast 1_A(n) \ll_\epsilon 1$ for all $n$?

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 $\chi(G) \leq \lceil \frac{1}{2}(\omega(G) + \Delta(G) + 1) \rceil.$

For every prime p, does the density of integers with h n = p exist?

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

Does liminf h n = ∞?

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 $K_{2n+1}$.

Is it true that if p is the greatest prime such that p - 1 ∣ n and p > n ^ ε, then h n = p?

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 $i$.

It is probably true that h n = 3 for infinitely many n.

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.

A Conjecture of Marian Deaconescu, see p.120 in https://doi.org/10.2307/2975810

[Needed to index shift in order to avoid trivial case $n = 0$, where the conjecture is trivially false.]

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

Let $\epsilon > 0$. Is there some set $A\subset\mathbb{N}$ of density $> 1 - \epsilon$ such that $a_1\cdots a_r = b_1\cdots b_s$ with $a_i, b_j\in A$ can only hold when $r = s$?