Open-problem statements, with their sources attached.

Binary-color case.* The $\gamma = 2$ specialization (two color classes).

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 $h(n)/n\to \infty$?

Infinite-target case.* When all $\kappa_\alpha \geq \aleph_0$ are infinite and bounded by $\lambda$, $\kappa_\alpha + 1 = \kappa_\alpha$, so the hypothesis simplifies to a "pure" stepping-down lemma:

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.

$r = 2$ case.* The stepping-down from 3-uniform to 2-uniform partition relations: $2^\lambda \to (\kappa_\alpha + 1)_{\alpha<\gamma}^3$ implies $\lambda \to (\kappa_\alpha)_{\alpha<\gamma}^2$. Generalises the classical Erdős–Rado stepping-up/down theorem for pairs.

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?

Let $\kappa$ be an uncountable cardinal. Must there exist a cardinal $\lambda$ such that every graph with chromatic number $\lambda$ contains a triangle-free subgraph with chromatic number $\kappa$?

Shelah proved that a negative answer is consistent when $\kappa = \lambda = \aleph_1$ (see erdos_1175.variants.shelah_consistency).

Threshold reformulation variant.* Replaces chromaticCardinal = λ in the hypothesis of erdos_1175 with λ ≤ chromaticCardinal (a graph of chromatic number ≥ λ has a triangle-free subgraph of chromatic number κ). This is a strengthening of erdos_1175 (see erdos_1175.test.threshold_implies_exact).

Does there exist, for all $r\geq 2$, a basis $A$ of order $r$ (so that $f_r(n)>0$ for all large $n$) such that $\sum_{n\leq x}f_r(n)^2 \ll x$ for all $x$?

For any $0<\alpha<1$, let $f(\alpha,n)=\frac{1}{\log n}\sum_{1\leq k\leq n}(\tfrac{1}{2}- \{ \alpha k\})$. Does $f(\alpha,n)$ have an asymptotic distribution function?

In other words, is there a non-decreasing function $g$ such that $g(-\infty)=0$, $g(\infty)=1$, and $\lim_{n\to \infty}\lvert \{ \alpha\in (0,1): f(\alpha,n)\leq c\}\rvert=g(c)$?

Is it true that in any 2-colouring of $\mathbb{N}$ there exists an infinite set $A$ such that all elements of $A+A$ are the same colour?

A conjecture of Owings [Ow74].

Are there infinitely many solutions to $\phi(n) = \phi(n+1)$, where $\phi$ is the Euler totient function?

Let $A \subseteq \mathbb{R}$ be an infinite set. Must there be a set $E \subseteq \mathbb{R}$ of positive measure which does not contain any set of the shape $a * A + b$ for some $a,b \in \mathbb{R}$ and $a \neq 0$?

Erdős [Er85e] says that, presumably, for every $k \geq 1$ the equation $\phi(n) = \phi(n+1) = \cdots = \phi (n+k)$ has infinitely many solutions.

[Er85e] Erdős, P., Some problems and results in number theory. Number theory and combinatorics. Japan 1984 (Tokyo, Okayama and Kyoto, 1984) (1985), 65-87.

Let G be a graph with n vertices such that every induced subgraph on ≥ $n/2$ vertices has more than $n^2/50$ edges. Must G contain a triangle?

For any fixed c > 0, if x is sufficiently large then there exists n ≤ x such that the values of φ(n+k) are all distinct for 1 ≤ k ≤ (log x)^c. This is an open problem.

A general version asks, for a fixed $r \in \mathbb{N}$, if a set $A \subseteq \{1, ..., N\}$ has no $a \in A$ and $b_1, ..., b_r \in A$ such that $a | (b_1 + ... + b_r)$ and $a < \min(b_1, ..., b_r)$, then is it true that $|A| \le N/(r+1) + O(1)$?

Let $t>1$ be a rational number. Is $\sum_{n=1}^\infty\frac{1}{t^n-1}=\sum_{n=1}^\infty \frac{\tau(n)}{t^n}$ irrational, where $\tau(n)$ counts the divisors of $n$?

A conjecture of Chowla.

Let $k≥3$. Are there $k$ consecutive primes in arithmetic progression?

Are there only finitely many unitary perfect numbers?