Open-problem statements, with their sources attached.

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

*The relation at $\omega_1$**: $\mathfrak{c} \to (\omega_1, n)^3_2$ for finite $n \ge 2$, where $\omega_1 = \aleph_1$ is the first uncountable ordinal.

Note that $\omega_1$ is not a countable ordinal, so this is not directly an instance of the main Erdős problem (which asks for countable $\beta$). Under CH, $\omega_1 = \mathfrak{c}.\mathrm{ord}$, making this a self-referential question about $\mathfrak{c}.\mathrm{ord} \to (\mathfrak{c}.\mathrm{ord}, n)^3_2$.

Granville and Soundararajan [GrSo98] have conjectured that at most $3$ powers of $2$ suffice for all odd integers, and hence at most $4$ powers of $2$ suffice for all even integers.

Ref: Granville, A. and Soundararajan, K., A Binary Additive Problem of Erdős and the Order of $2$ mod $p^2$

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?

Bogdan Grechuk has observed that $1117175146$ is not the sum of a prime and at most $3$ powers of $2$, and pointed out that parity considerations, coupled with the fact that there are many integers not the sum of a prime and $2$ powers of $2$ suggest that there exist infinitely many even integers which are not the sum of a prime and at most $3$ powers of $2$).

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?

Does every graph with chromatic number $\aleph_1$ contain a countable subgraph which is infinitely connected?

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

For every r ≥ 4 and k ≥ 2 is there some finite f(k,r) such that every graph of chromatic number ≥ f(k,r) contains a subgraph of girth ≥ r and chromatic number ≥ k?

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.

Are there infinitely many binomial coefficients with deficiency 1?

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

Are there only finitely many binomial coefficients with deficiency > 1?

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

Let $A=\{1\leq a_1 < a_2 < \cdots\}$ and $B=\{1\leq b_1 < b_2 < \cdots\}$ be sets of integers with $a_n/b_n\to 1$.

If $A+B$ contains all sufficiently large positive integers then is it true that $\limsup 1_A\ast 1_B(n)=\infty$?

A conjecture of Erdős and Sárközy.

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

Erdős Problem 1167.* Let $r \geq 2$ be finite, $\gamma \geq 2$, and $\lambda$ be an infinite cardinal. Let $\kappa_\alpha$ be cardinals for all $\alpha < \gamma$. Is it true that $2^\lambda \to (\kappa_\alpha + 1)_{\alpha < \gamma}^{r+1}$ implies $\lambda \to (\kappa_\alpha)_{\alpha < \gamma}^r?$ Here $+$ means cardinal addition, so that $\kappa_\alpha + 1 = \kappa_\alpha$ if $\kappa_\alpha$ is infinite.

A problem of Erdős, Hajnal, and Rado.

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

Finite-target case.* When all $\kappa_\alpha$ are finite, $\kappa_\alpha + 1$ is the ordinary natural-number successor. Special case of erdos_1167.

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