Open-problem statements, with their sources attached.

What is the largest product-free set in the alternating group $A_n$?

Assume for $n>1$, $f:\mathbb{R}^n\to\mathbb{R}^n$ is a bijection, where $\mathbb{R}^n$ is equipped with the standard topology. Does the connectedness of (the induced power set map) $f$ imply that of $f^{-1}$?

Does every almost-disjoint family of countably infinite sets whose pairwise intersections all have size ≠ 1 have Property B?

Formally: let α be any type, let (A_i)_{i ∈ I} be a family of countably infinite subsets of α such that for all i ≠ j, the intersection A_i ∩ A_j is finite and |A_i ∩ A_j| ≠ 1. Does there exist a 2-colouring f : α → Fin 2 such that no A_i is monochromatic?

This is an open question about Property B for almost-disjoint families with a forbidden intersection size of 1. Note:* This generalises the formulation in which the ground set is ℕ. Since every countably infinite set is in bijection with ℕ, the two formulations are equivalent, but working over an arbitrary ground type makes the statement apply immediately to, e.g., almost-disjoint families of countable subsets of an uncountable space.

If f(z) = ∑ aₖzⁿₖ is an entire function (with aₖ ≠ 0 for all k) such that nₖ / k → ∞, is it true that f assumes every value infinitely often?

Is $e \pi$ irrational?

(D) Breakdown of Navier–Stokes Solutions on ℝ³/ℤ³.

The density of hyperbolicity conjecture, stating that the set of all parameters c for which fun z ↦ z ^ 2 + c has an attracting cycle is dense in the Mandelbrot set.

Problem 4.3.* Let $\Omega \subset \mathbb{R}$ be a finite union of intervals and $\nu$ a weak tiling measure for $\Omega$. Must $\nu$ be expressible as a convex combination of proper tiling measures?

For $n > 8$, $2^n$ is not the the sum of distinct powers of $3$. Expressed here in terms of the base $3$ digits of $n$.

This conjecture is equivalent to the halting of a $15$-state $2$-symbol Turing Machine.

TODO(lezeau): Formalize the Turing Machine version of this problem.

Source: Hardness of Busy Beaver Value BB(15): https://link.springer.com/chapter/10.1007/978-3-031-72621-7_9 This is also https://arxiv.org/abs/2107.12475.

Let $\alpha(n)$ be such that every set of $n$ points in the unit disk contains three points which determine a triangle of area at most $\alpha(n)$. Estimate $\alpha(n)$.

Let $f(n)$ be minimal such that any $f(n)$ points in $ℝ^2$, no three on a line, contain $n$ points which form the vertices of a convex $n$-gon. Prove that $f(n) = 2^{n-2} + 1$.

Problem 13 in [Ar2013]: Is it true that every infinite homogeneous compact hausdorff space contains a non-trivial convergent sequence?

Černý Conjecture*: Every synchronizing DFA with $n$ states admits a synchronizing word of length at most $(n - 1)^2$.

Conjecture 4.2*: For any prime $k$ larger than $233$, $S(k) > 2k$.

Let $f(z)=\sum_{0\leq k\leq n} \epsilon_k z^k$ be a random polynomial, where $\epsilon_k\in \{-1,1\}$ independently uniformly at random for $0\leq k\leq n$.

Is it true that, if $R_n$ is the number of roots of $f(z)$ in $\{ z\in \mathbb{C} : \lvert z\rvert \leq 1\}$, then

almost surely?

There is some ambiguity as to whether the intended coefficient set is $\{-1, 1\}$ or $\{0, 1\}$, see erdos_522.variants.zero_one for the alternate version.

Benchmark open subproblem: existence of a SIC-POVM in dimension $64$.

The Köthe conjecture: for any nil ideal I of R, the matrix ideal M_n(I) is a nil ideal of the matrix ring M_n(R).

Let $G$ and $H$ be finite groups of the same order with $\sum_{g \in G} \phi(|g|) = \sum_{h \in H} \phi(|h|)$, where $\phi$ is the Euler totient function. Suppose that $G$ is simple. Is $H$ necessarily simple?

Let $f : \mathbb R^n \to \mathbb R, n \geq 2$ be a $C^1$ function. Does the equality $\sup_{x \in \mathbb R^n}f(x) = \sup_{x\in \mathbb R^n} f(x+\nabla f(x))$ hold when both suprema are finite?

Let $X$ be a set of cardinality $\aleph_\omega$ and $f$ be a function from the finite subsets of $X$ to $X$ such that $f(A)\not\in A$ for all $A$. Must there exist an infinite $Y\subseteq X$ that is independent - that is, for all finite $B\subset Y$ we have $f(B)\not\in Y$?