Open-problem statements, with their sources attached.

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

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

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:

$r = 2$ 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.

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 $x$?

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].

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 $a \neq 0$?

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?

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

Are there $11$ consecutive primes in arithmetic progression?

It is open, even for $k=3$, whether there are infinitely many such progressions.

Fix a $k \geq 3$. Is it true that there are infinitely many arithmetic prime progressions of length $k$?

Let $A$ be a finite Sidon set and $A+A=\{s_1<\cdots<s_t\}$. Is it true that $\frac{1}{t}\sum_{1\leq i<t}(s_{i+1}-s_i)^2 \to \infty$ as $\lvert A\rvert\to \infty$?

Is it true that for every $k \geq 1$ we have

for all sufficiently large $N$?

Does there exist a maximal Sidon set $A\subset \{1,\ldots,N\}$ of size $O(N^{1/3})$?

A question of Erdős, Sárközy, and Sós [ESS94].

Let A be an infinite B₂[2] set. Must liminf |A ∩ {1, ..., N}| * N ^ (- 1 / 2) = 0?

Estimate $h(n)$ by finding a better upper bound.