Open-problem statements, with their sources attached.

For $N = 6$ and $D = 4$, does there exist no solution to the monochromatic quantum graph equation system over $\mathbb{C}$?

Let $\tau(n)$ count the number of divisors of $n$. Is there some $n > 24$ such that

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

Erdős says 'it is extremely doubtful' that there are infinitely many such $n$, and in fact suggests that

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

Erdős says it 'seems certain' that for every $k$ there are infinitely many $n$ for which

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

Is there and $A \subset \mathbb{N}$ is such that $\lim_{n\to \infty}\frac{1_A\ast 1_A(n)}{\log n}$ exists and is $\ne 0$?

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

Can the product of an arithmetic progression of positive integers $n, n + d, ..., n + (k - 1)d$ of length ≥ 4, with $(n, d) = 1$, be a perfect power?

For $N = 10$ and $D = 4$, does there exist no solution to the monochromatic quantum graph equation system over $\mathbb{C}$?

Denote by $M(n, k)$ the least common multiple of the finite set $\{n+1, \dotsc, n+k\}$. Is it true that for all $m \geq n + k$, we get $M(m, k) \neq M(n, k)$?

For $N = 10$ and $D = 5$, does there exist no solution to the monochromatic quantum graph equation system over $\mathbb{C}$?

Is $\sum_{n=2}^\infty \frac{1}{n!-1}$ irrational?

For $N = 10$ and $D = 6$, does there exist no solution to the monochromatic quantum graph equation system over $\mathbb{C}$?

Is it true that, for all sufficiently large $n$, there exists some $k$ such that

where $p(m)$ denotes the least prime factor of $m$?

For $N = 10$ and $D = 7$, does there exist no solution to the monochromatic quantum graph equation system over $\mathbb{C}$?

Can one prove this is false if we replace $k^2+1$ by $e^{(1+\epsilon)\sqrt{k}}+C_\epsilon$, for all $\epsilon>0$, where $C_\epsilon>0$ is some constant?

For $N = 10$ and $D = 8$, does there exist no solution to the monochromatic quantum graph equation system over $\mathbb{C}$?

Erdős problem 681.* Is it true that for all large $n$ there exists $k$ such that $n + k$ is composite and $p(n+k) > k^2$, where $p(m)$ is the least prime factor of $m$ ?