Open-problem statements, with their sources attached.

Is there some constant $c > 0$ such that, for all large enough $n$ and all polynomials $P$ of degree $n$ with coefficients in $\{-1, 1\}$, $\max_{|z|=1} |P(z)| > (1 + c) \sqrt{n}?$

If $G$ is a group, can there exist an exact covering of $G$ by more than one coset of different sizes? (i.e. each element is contained in exactly one of the cosets.)

The conjectured answer is no: in every such exact covering, two of the subgroups have the same cardinality.

Let $C>0$. There exists $\epsilon>0$ such that if $n$ is sufficiently large the following holds.

For any $x_1,\ldots,x_n\in [-1,1]$ there exist $y_1,\ldots,y_n\in [-1,1]$ such that, if $P$ is a polynomial of degree $m<(1+\epsilon)n$ with $P(x_i)=y_i$ for at least $(1-\epsilon)n$ many $1\leq i\leq n$, then $\max_{x\in [-1,1]}\lvert P(x)\rvert >C.$

What is the infimum of |{x ∈ ℝ : |f x| < 1}| over all nonconstant monic polynomials f such that all of its roots are real and contained in [-1,1]?

Let $G$ be a graph with chromatic number $\aleph_1$. Is it true that there is a colouring of the edges with $\aleph_1$ many colours such that, in any countable colouring of the vertices, there exists a vertex colour containing all edge colours?

A problem of Erdős, Galvin, and Hajnal. The consistency of this was proved by Hajnal and Komjáth.

Let $f(z) ∈ ℂ[z]$ be a monic non-constant polynomial. Can the set $\{z ∈ ℂ : |f(z)| ≤ 1\}$ be covered by a set of closed discs the sum of whose radii is $≤ 2$?

Let $f(z) = \prod_{i=1}^{n} (z - z_i) \in \mathbb{C}[x]$ with $|z_i| < 1$ for all $i$.

Conjecture: Must there always exist a path of length less than 2 in $\{ z \in \mathbb{C} \mid |f(z)| < 1 \}$ which connects two of the roots of $f$?

Let $a_1 < a_2 < \dots$ be a sequence of integers such that $\lim_{n\to\infty} \frac{a_n}{a_{n-1}^2} = 1$ and $\sum \frac{1}{a_n} \in \mathbb{Q}$.

Then, for all sufficiently large $n \ge 1$, $a_n = a_{n-1}^2 - a_{n-1} + 1$.

Does there exists a positive constant C such that for all f ∈ L²[0,1] and all lacunary sequences n, if ‖f - fₖ‖₂ = O(1 / log log log k ^ C), then for almost every x, lim ∑ k ∈ Finset.range N, f (n k • x)) / N = ∫ t, f t ∂t?

Let $A\subset \mathbb{R}^2$ be a set of $n$ points with no three on a line. Does $A$ determine at least $\lfloor n/2\rfloor$ distinct distances?

Is the diameter of $A$ at least $Cn$ for some constant $C > 0$?

Is there a polynomial $f:\mathbb{Z}\to \mathbb{Z}$ of degree at least $2$ and a set $A\subset \mathbb{Z}$ such that for any $z\in \mathbb{Z}$ there is exactly one $a\in A$ and $b\in \{ f(n) : n\in\mathbb{Z}\}$ such that $z=a+b$?

Let $G$ be a group, and let $A = \{a_1G_1, \dots, a_kG_k\}$ be a finite system of left cosets of subgroups $G_1, \dots, G_k$ of $G$.

Herzog and Schönheim conjectured that if $A$ forms a partition of $G$ with $k > 1$, then the indices $[G:G_1], \dots, [G:G_k]$ cannot be distinct.

Determine which countable ordinals $β$ have the property that, if $α = \omega^β$, then in any red/blue colouring of the edges of $K_α$ there is either a red $K_α$ or a blue $K_3$.

Let f be a transcendental entire function. What is the greatest possible value of liminf (fun r : ℝ => ratio r f) atTop?

Is there some $c > 0$ such that every measurable $A \subseteq \mathbb{R}^2$ of measure $\geq c$ contains the vertices of a triangle of area 1?

Stronger conjecture: diameter $\geq n - 1$ for sufficiently large $n$.

Probably there is no such $A$ for the polynomial $X^3$.

Erdős Problem 598:* Let $m$ be an infinite cardinal and $\kappa$ be the successor cardinal of $2^{\aleph_0}$. Can one colour the countable subsets of $m$ using $\kappa$ many colours so that every $X \subseteq m$ with $|X| = \kappa$ contains subsets of all possible colours?

Is it true that for all entire functions f = ∑ aₖzⁿₖ such that ∑' 1 / nₖ < ∞, limsup (fun r => ratio r f) atTop = 1?