Open-problem statements, with their sources attached.

Brennan's conjecture, part 2: $B_b(-2) = 1$.

Is the Euler-Mascheroni constant $\gamma$ irrational?

De Giorgi's conjecture holds in dimension $n = 7$.

Bogdan Grechuk has observed that $1117175146$ is not the sum of a prime and at most $3$ powers of $2$, and pointed out that parity considerations, coupled with the fact that there are many integers not the sum of a prime and $2$ powers of $2$ suggest that there exist infinitely many even integers which are not the sum of a prime and at most $3$ powers of $2$).

Is $\varepsilon^{-C}$ rotations enough?

The Hadwiger–Nelson problem asks: How many colors are required to color the plane such that no two points at distance 1 from each other have the same color?

Problem 2 in [Ar2013]: Give an example in ZFC of a weakly first- countable compact Hausdorff space X such that $𝔠 < |X|$.

Note: [Ar2013] uses a blanket convention that all spaces are Tychonoff and "compact" means compact Hausdorff.

Problem 10.2. To prove that $\lVert e^n \rVert$ does not tend to 0 as n tends to infinity.

The second Cuboid conjecture

Benchmark open subproblem: existence of a SIC-POVM in dimension $72$.

Conjecture:* Are there infinitely many Leinster groups?

This asks whether there exist infinitely many (non-isomorphic) finite groups that are Leinster groups.

Formalized via the negation of "Does there exist an n such that all Leinster groups have order less than n".

The Vaught conjecture states that for a countable language L and a complete L-Theory T the number of countable models of T (up to isomorphism) is finite, $\aleph_0$ or $2^{\aleph_0}$.

The Gompertz constant $\delta$ is transcendental.

De Giorgi's conjecture holds in dimension $n = 8$.

Does every graph with chromatic number $\aleph_1$ contain a countable subgraph which is infinitely connected?

Can the Cohn-Elkies scheme be used to prove the optimal bound for circle-packings in 2 dimensions?

Erdős [Er46] asked whether every set of $n$ distinct points in $\mathbb{R}^2$ determines $\gg \frac{n}{\sqrt{\log n}}$ many distinct distances.

Problem 3 in [Ar2013]: Give an example in ZFC of a weakly first- countable compact Hausdorff space which is not first countable.

Note: [Ar2013] uses a blanket convention that all spaces are Tychonoff and "compact" means compact Hausdorff.

Problem 10.3. To prove that there exists a positive real number~$c$ such that $\lVert e^n \rVert > e^{−cn}$, for every~$n \ge 1$. Posed by Mahler [Mah53].

The third Cuboid conjecture