Open-problem statements, with their sources attached.

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 $N=5^n$?

Conjecture 7 from Kahn–Kalai 2006: the same statement as the original conjecture, but with the additional assumption that $t$ is the critical probability for $F$, namely $\mu_t(F) = 1/2$.

Special case in dimension $6$: determine the maximal number of mutually unbiased orthonormal bases in $\mathbb{C}^6$.

Special case in dimension $10$ (not a prime power): determine the maximal number of mutually unbiased orthonormal bases in $\mathbb{C}^{10}$.

Special case in dimension $12$ (not a prime power): determine the maximal number of mutually unbiased orthonormal bases in $\mathbb{C}^{12}$.

Special case in dimension $14$ (not a prime power): determine the maximal number of mutually unbiased orthonormal bases in $\mathbb{C}^{14}$.

Special case in dimension $15$ (not a prime power): determine the maximal number of mutually unbiased orthonormal bases in $\mathbb{C}^{15}$.

Open Quantum Problem 13: determine the maximal number of mutually unbiased orthonormal bases in $\mathbb{C}^d$ for $d \ge 2$.

Open benchmark statement: does an $\mathrm{AME}(7,6)$ state exist?

Open benchmark statement: does an $\mathrm{AME}(7,10)$ state exist?