Open-problem statements, with their sources attached.

Let $k \ge 4$ and $r\ge 1$. Must there exist a graph $G$ with chromatic number $k$ such that every vertex is critical, yet every critical set of edges has size $>r$?

Let $k \ge 4$. Must there exist a graph $G$ with chromatic number $k$ such that every vertex is critical, yet every critical set of edges has size $>1$?

This was conjectured by Dirac in 1970.

The case $k=4$ and $r=1$ remains open: Are there $4$-critical graphs without any critical edges?

Is it true that $F(x) \leq (\log x)^{O(1)}$?

Is there a constant $C > 0$ such that, for all large $x$, every interval $[x, x+(\log x)C]$ contains two integers with the same number of divisors?

Is it true that $\liminf f(n)=1$?

Is it true that $\limsup f(n)=\infty$?

Is it true that $f(n)=o(\log\log n)$ for all $n$?

Erdős writes that a 'weaker conjecture which is perhaps not quite inaccessible' is that, for every $\epsilon>0$, if $x$ is sufficiently large there exists $y<x$ such that $\pi(x)< \pi(y)+\epsilon \pi(x-y)$. Compare this to [855].

The study of $f(p)$ is even harder, and Erdős could not prove that $\sum_{p<x}f(p)^2\sim \pi(x)$.

If 1 < a 0 < ... has property Erdos951Prop, is it true that #{a i ≤ x} ≤ π x?

Is there an infinite sequence of distinct Gaussian primes $x_1,x_2,\ldots$ such that $\lvert x_{n+1}-x_n\rvert \ll 1$?

It is conjectured that $f(k) \ll (\log k)^O(1)$.

Main conjecture:

$\log k(n) \le (\log n)^{(1/2 + o(1))}$

Does the set {n | u n < u (n+1)} have positive natural density?

Erdős asked whether there are infinitely many solutions to uₙ < uₙ₊₁ < uₙ₊₂.

Erdős asked whether there are infinitely many solutions to uₙ > uₙ₊₁ > uₙ₊₂.

Let p(a, d) be the least prime congruent to a (mod d). Does there exist a constant c > 0 such that for all large d, p(a, d) > (1 + c) * φ(d) * log d for ≫ φ(d) many values of a?

Erdős problem 972.* Let $\alpha > 1$ be irrational. Are there infinitely many primes $p$ such that $\lfloor p\alpha \rfloor$ is also prime?

Does there exist a constant $C>1$ such that, for every $n\geq 2$, there exists a sequence $z_i\in \mathbb{C}$ with $z_1=1$ and $\lvert z_i\rvert \geq 1$ for all $1\leq i\leq n$ with $\max_{2\leq k\leq n+1}\left\lvert \sum_{1\leq i\leq n}z_i^k\right\rvert < C^{-n}$?

This is Problem 7.3 in [Ha74], where it is attributed to Erdős.