Open-problem statements, with their sources attached.

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

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

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 ${}^2G_2(q)$, $q = 3^{2m+1}$.

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

Let $A \subset \mathbb{F}_2^n$ be a set of density $\alpha > 0$. Does $10A$ contain a coset of some subspace of dimension at least $n - O(\log(1/\alpha))$?

More precisely: does there exist an absolute constant $C > 0$ such that for all $n \geq 1$ and all nonempty $A \subseteq \mathbb{F}_2^n$ with density $\alpha > 0$, the sumset $10A$ contains a coset of some subspace of dimension at least $n - C \log_2(1/\alpha)$?

The sumset $10A$ is defined as $\{a_1 + a_2 + \cdots + a_{10} : a_i \in A\}$, using the pointwise scalar multiplication notation 10 • A where • denotes the iterated addition of a set.

Note: We model $\mathbb{F}_2^n$ as Fin n → ZMod 2, which is an $n$-dimensional vector space over $\mathbb{F}_2$.

Suppose that $A \subset \mathbb{F}_2^n$ is a set of density $\alpha$. What is the largest size of coset guaranteed to be contained in $2A$?

We phrase this by asking for the exact function $F(\alpha, n)$ giving the maximum dimension of a guaranteed coset.

Suppose that $A \subset \mathbb{F}_2^n$ has density $\alpha > 1/2 - C/\sqrt{n}$. Does $A + A$ contain a subspace of co-dimension $O_C(1)$? [Sa11, Question 5.1]

Suppose that $A \subset \mathbb{F}_2^n$ is a set with an additive complement of size $K$. Does $2A$ contain a coset of codimension $O_K(1)$?

Could $2A$ even contain a coset of codimension $O(\log K)$?

Suppose $A, B ⊆ \{1, \dots, N\}$ both have size at least $N^{0.49}$. Must the sumset $A + B$ contain a composite number?

The no-k-in-line problem: For $N \geq k$ and $k > 2$, the AllowedSetSize is $(k - 1) N$, i. e. on an $N \times N$ subset, there is a set of $(k - 1) N$ points for which no $k$ lie on a line (and not such a set of bigger size).

Green's Open Problem 72 / No-three-in-line problem*: The no-k-in-line conjecture holds for $k = 3$.

Does the no-three-in-line problem hold when $N$ is big enough?

Given $n$ points in the unit disc, must there be a triangle of area at most $n^{-2+o(1)}$ determined by them?

Problem 9 (ii): is $r_5(N) \ll N(\log N)^{-c}$?

Problem 9 (iii): is $r_4(\mathbf{F}_5^n) \ll N^{1-c}$, where $N=5^n$?

Let $A \subset \mathbf{Z}$ be a set of $n$ integers. Is there a set $S \subset A$ of size $(\log n)^{100}$ such that the restricted sumset$S \hat{+} S$ is disjoint from $A$?

Suppose that $A \subset [0,1]$ is open and has measure greater than $\frac{1}{3}$. Is there a solution to $xy = z$ with $x, y, z \in A$?

Sieve $[N]$ by removing half the residue classes mod $p_i$, for primes $2 \leqslant p_1 < p_2 < \dots < p_{1000} < N^{9/10}$. Does the remaining set have size at most $\frac{1}{10} N$?

We interpret "half the residue classes" as $\lfloor p_i / 2 \rfloor$.

We conjecture that the best-known lower bound can be improved.