Open-problem statements, with their sources attached.

If $A \subset \mathbb{Z}/p\mathbb{Z}$ is random, $|A| = \sqrt{p}$, can we almost surely cover $\mathbb{Z}/p\mathbb{Z}$ with $100\sqrt{p}$ translates of $A$? [Gr24]

Can $\binom{n}{2}$ be the product of consecutive primes infinitely often?

"I do not know how to answer this even with 100 replaced by 1.01." [Gr24]"

The following is Schinzel's conjecture, which appears in [Gu04].

Similar questions are interesting with $\sqrt{p}$ replaced by $p^\theta$ for any $\theta \le 1/2$. [Gr24]

NOTE: using $C p^\theta$ translates as stated makes the conjecture trivially false by the pigeonhole principle. Indeed for a set of size $p^\theta$, we cover at most $C p^{2\theta}$ elements, which is strictly less than $p$ for $\theta < 1/2$. We interpret the question as asking whether $O(p^{1-\theta})$ translates suffice. This generalizes the main conjecture where $\sqrt{p} = p^{1-1/2}$.

Is it true that for every $n \geq 1$ there is a $k$ such that

Does $f(r) \to \infty$? [Gr24]

Is there an infinite Sidon set $A\subset \mathbb{N}$ such that $\lvert A\cap \{1\ldots,N\}\rvert \gg_\epsilon N^{1/2-\epsilon}$ for all $\varepsilon > 0$?

The possibility that f(r) = 1 for all r has not been ruled out [Gr24]

Does there exists a constant c such that f n - 2 * n ~ c * (n / log n)?

It is not known whether f(2) = 1 [Gr24]

Is it true that $\sum_{n\leq x}t_2(n)\ll \frac{x^2}{(\log x)^c}$ for some $c>0$?

Does $\tilde{f}(r) \to \infty$? [Gr24]

Is it true that, for $k\geq 2$, $\sum_{n\leq x}t_{k+1}(n) =o\left(\sum_{n\leq x}t_k(n)\right)?$

Does $f_{\text{all}}(r) \to \infty$? [Gr24]

Erdős and Hall conjecture that the sum is $o(x^2/(\log x)^c)$ for any $c<\log 2$.

Which finite groups have the smallest biggest product-free sets?

We formalise this as: determine the supremum of exponents $\alpha$ such that every nontrivial finite group of order $n$ contains a product-free set of size $\geq c n^{\alpha}$ for some absolute constant $c > 0$. (The trivial group is excluded since its only product-free subset is empty.) Kedlaya [Ke97] showed that $\alpha = 11/14$ is admissible, and Green suggests this exponent may well be sharp; the candidate extremal family is the Ree groups ${}^2G_2(q)$, $q = 3^{2m+1}$.

They ask about the behaviour of $t_{n-3}(n!)$ and also ask whether, for infinitely many $n$, $t_k(n!)< t_{k-1}(n!)-1$ for all $1\leq k < n$.

A good model problem would be to determine the largest product-free subsets of $\mathrm{SL}_2(\mathbb{F}_p)$.

Is it true that for every $k$ there exists $n$ such that $\prod_{0\leq i\leq k}(n-i) \mid \binom{2n}{n}?$