Open-problem statements, with their sources attached.

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

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?.

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

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?

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?

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 $c > 0$?

If $n$ points in $\mathbb{R}^2$ form a convex polygon then there are $O(n)$ many pairs which are distance $1$ apart.

Does every convex polygon have a vertex with no other 4 vertices equidistant from it?

Erdős also conjectured that there is a $k$ for which every convex polygon has a vertex with no other $k$ vertices equidistant from it.

Let $h(n)$ be such that any $n$ points in $\mathbb{R}^2$, with no three on a line and no four on a circle, determine at least $h(n)$ distinct distances. Does $h(n)/n\to \infty$?

If $n$ distinct points in $\mathbb{R}^2$ form a convex polygon then some vertex has at least $\lfloor\frac{n}{2}\rfloor$ different distances to other vertices.

For sufficiently large n, is it the case that any set of n points with minimum distance $1$ that minimizes diameter must contain an equilateral triangle of side length 1?

Moser's Worm Problem* What is the minimal area (or greatest lower bound on the area) of a shape that can cover every unit-length curve?