Open-problem statements, with their sources attached.

Does the limit $\lim_{n\to\infty} \frac{f(2n)}{f(n)}$ tend to infinity?

(Other finite limits have been ruled out by [KoLu25], see below)

Let $\frac a b\in \mathbb{Q}_{>0}$ with $b$ squarefree. Are there integers $1 < n_1 < \dots < n_k$, each the product of two distinct primes, such that $\frac{a}{b}=\frac{1}{n_1}+\cdots+\frac{1}{n_k}$?

Is the upper density of the set of odd numbers that cannot be expressed as a prime plus two powers of 2 positive?

Are there two finite set of primes $P$ and $Q$ such that

?

Asked by Barbeau [Ba76].

[Ba76] Barbeau, E. J., Computer challenge corner: Problem 477: A brute force program.

Is there a graph with $\aleph_2$ vertices and chromatic number $\aleph_2$ such that every subgraph on $\aleph_1$ vertices has chromatic number $\leq\aleph_0$?

Are there infinitely many pairs (m, P) where m ≥ 2 is an integer and P is a set of distinct primes such that the following equation holds: $\sum_{p \in P} \frac{1}{p} = 1 - \frac{1}{m}$?

Is there a graph with $\aleph_{\omega+1}$ vertices and chromatic number $\aleph_1$ such that every subgraph on $\aleph_\omega$ vertices has chromatic number $\leq\aleph_0$?

It is conjectured that the set of primary pseudoperfect numbers is infinite.

Is there a graph with $\aleph_2$ vertices and chromatic number $\aleph_2$ such that every subgraph on $\aleph_1$ vertices has chromatic number $\leq\aleph_0$?

Is there some constant $c>0$ such that for every $n\geq 1$ there exists some $\delta_k\in \{-1,0,1\}$ for $1\leq k\leq n$ with $0< \left\lvert \sum_{1\leq k\leq n}\frac{\delta_k}{k}\right\rvert < \frac{c}{2^n}?$

Is there a graph with $\aleph_{\omega+1}$ vertices and chromatic number $\aleph_1$ such that every subgraph on $\aleph_\omega$ vertices has chromatic number $\leq\aleph_0$?

Is it true that for sufficiently large $n$, for any $\delta_k\in \{-1,0,1\}$, $\left\lvert \sum_{1\leq k\leq n}\frac{\delta_k}{k}\right\rvert > \frac{1}{[1,\ldots,n]}$ whenever the left-hand side is not zero?

Is it true that, for $k\geq 4$, $f_k(n) \gg \frac{n^{1-\frac{1}{k-1}}}{(\log n)^{c_k}}$ for some constant $c_k>0$?

Does there exist a set $A \subseteq \mathbb{N}$ such that $|A \cap \{1, \ldots, N\}| = o((\log N)^2)$ and every sufficiently large integer can be written as $p + a$ for some prime $p$ and $a \in A$?

Let $S \subseteq \mathbb{R}$ be a set containing no solutions to $a + b = c$. Must there be a set $A \subseteq \mathbb{R} \setminus S$ of cardinality continuum such that $A + A \subseteq \mathbb{R}\setminus S$?

Can the bound $O(\log N)$ be achieved for an additive complement to the primes? [Guy04] writes that Erdős offered $50 for the solution.

Let $A$ be a set of $n$ positive integers. Does $A$ contain a sum-free set of size at least $\frac n 3 + Ω(n)$, where $Ω(n) → ∞$ as $n → ∞$?

Let $R(N)$ be the size of the largest $A\subseteq\{1, ..., N\}$ such that all sums $\sum_{n\in S} \frac{1}{n}$ are distinct for $S\subseteq A$. What is $R(N)$?

Let $G$ be an abelian group of size $N$, and suppose that $A \subset G$ has density $\alpha$. Are there at least $\alpha^{15} N^{10}$ tuples $(x_1, \dots, x_5, y_1, \dots, y_5) \in G^{10}$ such that $x_i + y_j \in A$ whenever $j \in \{i, i+1, i+2\}$?

Note: We interpret indices modulo 5.

Let $R(N)$ be the size of the largest $A\subseteq\{1, ..., N\}$ such that all sums $\sum_{n\in S} \frac{1}{n}$ are distinct for $S\subseteq A$. What is $\Theta(R(N))$?