Open-problem statements, with their sources attached.

Is the diameter of $A$ at least $Cn$ for some constant $C > 0$?

Suppose that $A$ is an open subset of $[0, 1]^2$ with measure $\alpha$. Are there four points in $A$ determining an axis-parallel rectangle with area $\gt c \alpha^2$?

Let $K \subset \mathbb{R}^n$ be a balanced compact set (that is, $\lambda K \subseteq K$ whenever $|\lambda| \leq 1$) and suppose that the normalised Gaussian measure $\gamma_n(K) \geq 0.99$. Does $10K$ contain a compact convex set $C$ with $\gamma_n(C) \geq 0.01$?

Stronger conjecture: diameter $\geq n - 1$ for sufficiently large $n$.

Given $n$ points in $\mathbb{R}^2$, no five of which are on a line, the number of lines containing four points is $o(n^2)$.

Let $f(n)$ be minimal such that any $f(n)$ points in $ℝ^2$, no three on a line, contain $n$ points which form the vertices of a convex $n$-gon. Prove that $f(n) = 2^{n-2} + 1$.

Erdős conjectured that the triangular lattice is best possible in 2D, in particular that $f_2(3n^2 + 3n + 1) < 9n^2 + 3n$.

Note: in [Er75f] is read $9n^2 + 6n$, but this seems to be a typo.

Is the $n^{4/3}\log\log n$ lower bound in 3D also an upper bound?.

How many rotated (about the origin) copies of the 'pyjama set' $\\{(x, y) \in \mathbb{R}^2 : \text{dist}(x, \mathbb{Z}) \leq \varepsilon\\}$ are needed to cover $\mathbb{R}^2$?

In particular, can one find a better bound than the best-known bound from [KrLe25]?

Is there a dense subset of ℝ^2 such that all pairwise distances are rational?

Is there a better bound than the best-known bound from [KrLe25]? This is an existential version of the main problem that does not require providing the bound explicitly.

Let $n \geq 4$. Are there $n$ points in $\mathbb{R}^2$, no three on a line and no four on a circle, such that all pairwise distances are integers?

Is $\varepsilon^{-C}$ rotations enough?

The Hadwiger–Nelson problem asks: How many colors are required to color the plane such that no two points at distance 1 from each other have the same color?

Can the Cohn-Elkies scheme be used to prove the optimal bound for circle-packings in 2 dimensions?

Erdős [Er46] asked whether every set of $n$ distinct points in $\mathbb{R}^2$ determines $\gg \frac{n}{\sqrt{\log n}}$ many distinct distances.

Suppose $A\subset \mathbb{R}^2$ has $\lvert A\rvert=n$ and minimises the number of distinct distances between points in $A$. Prove that for large $n$ there are at least two (and probably many) such $A$ which are non-similar.

Is it true that $f(n)\leq n^{o(1)}$?

Or even $f(n) < n^{c/\log\log n}$ for some constant $c > 0$?

Sendov's conjecture* states that for a polynomial $f(z)=(z-r_{1})\cdots (z-r_{n}),\qquad (n\geq 2)$ with all roots $r_1, ..., r_n$ inside the closed unit disk $|z| ≤ 1$, each of the $n$ roots is at a distance no more than $1$ from at least one critical point.