Open-problem statements, with their sources attached.

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:

What is the smallest square that can contain 21 unit circles?

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.

Problem 10.8 ($p$-adic Littlewood conjecture). For every real number $\xi$ and every prime number $p$, $\inf_{q \ge 1} q \cdot \lVert q \xi \rVert \cdot |q|_p = 0,$ where $\lVert \cdot \rVert$ denotes the distance to the nearest integer and $|\cdot|_p$ denotes the $p$-adic absolute value. Posed by de Mathan and Teulié [dMT04].

$r = 2$ 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.

What is the smallest circle that can contain 15 unit circles?

Reference: Graham RL, Lubachevsky BD, Nurmela KJ, Ostergard PRJ. Dense packings of congruent circles in a circle. Discrete Math 1998;181:139–154.

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?

The sequence (3/2)^n is equidistributed modulo 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).

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?

For any transcendental number x, the sequence x * (3 / 2) ^ n is equidistributed modulo 1.

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).

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

Find an accumulation point of the sequence (3/2)^n modulo 1.

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.