Open-problem statements, with their sources attached.

Let $r\geq 3$. If the edges of $K_{r^2+1}$ are $r$-coloured then there exist $r+1$ vertices with at least one colour missing on the edges of the induced $K_{r+1}$.

In other words, there is no balanced colouring.

A conjecture of Erdős and Gyárfás [ErGy99].

A conjecture by Heath-Brown: The sum of squares of the first $N$ gaps between consecutive primes behaves like $N * (log N)^2$.

Let $X$ be a finite set of size $n$ and $H(n)$ be such that there is a function $f:\{A : A\subseteq X\}\to X$ so that for every $Y\subseteq X$ with $\lvert Y\rvert \geq H(n)$ we have $\left\{ f(A) : A\subseteq Y\right\}=X$. Prove that $H(n)-\log_2 n \to \infty$.

Is it true that for all c ≥ 0, the density f c of integers for which (p (n + 1) - p n) / log n < c exists and is a continuous function of c?

Which triangles can only be decomposed into a square number of congruent triangles?

Let c₁, c₂ > 0. Is it true that for any sufficiently large x, there exists more than c₁ * log x many consecutive primes ≤ x such that the difference between any two is > c₂?

Does every finite graph with minimum degree at least $3$ contain a cycle of length $2^k$ for some $k \geq 2$?

For every $n>2$ there exist distinct integers $1 ≤ x < y < z$ such that $\frac 4 n = \frac 1 x + \frac 1 y + \frac 1 z$.

Let $x_1,\ldots,x_n\in \mathbb{R}^2$ and let $R(x_i)=\#\{ \lvert x_j-x_i\rvert : j\neq i\}$, where the points are ordered such that $R(x_1)\leq \cdots \leq R(x_n).$ Let $g(n)$ be the maximum number of distinct values the $R(x_i)$ can take. Is it true that $g(n) \geq (1-o(1))n$?

Schinzel conjectured (see [Si56]) the generalisation that, for any fixed $a$, if $n$ is sufficiently large in terms of $a$ then there exist distinct integers $1\leq x < y < z$ such that $\frac{a}{n} = \frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$

Let $x_1,\ldots,x_n\in \mathbb{R}^2$ be such that no circle whose centre is one of the $x_i$ contains three other points. Are there at least$(1+c)\frac{n}{2}$ distinct distances determined between the $x_i$, for some constant $c>0$ and all $n$ sufficiently large?

In the spirit of related conjectures of Erdős and others, presumably some kind of assumption that the points are in general position (e.g. no three on a line and no four on a circle) was intended.

Let $C > 1$. Does the set of integers of the form $p + \lfloor C^k \rfloor$, for some prime $p$ and $k\geq 0$, have density $>0$?

Let $\mathcal{F}$ be a family of sets closed under taking subsets (i.e. if $B\subseteq A\in\mathcal{F}$ then $B\in \mathcal{F}$). There exists some element $x$ such that whenever $\mathcal{F}'\subseteq \mathcal{F}$ is an intersecting subfamily we have $\lvert \mathcal{F}'\rvert \leq \lvert \{ A\in \mathcal{F} : x\in A\}\rvert.$

Let $n_1 < n_2 < \cdots$ be a sequence of integers such that

Is

transcendental?

If there is a finite projective plane of order $n$ then must $n$ be a prime power?

Is $\sum_{n} \frac{\phi(n)}{2^n}$ irrational? Here $\phi$ is the Euler totient function.

It is open whether there exists a projective plane of order 12.

Let $n_1 < n_2 < \dots$ be an arbitrary sequence of integers, each with an associated residue class $a_i \pmod{n_i}$. Let $A$ be the set of integers $n$ such that for every $i$ either $n < n_i$ or $n \not\equiv a_i \pmod{n_i}$. Must the logarithmic density of $A$ exist?

Let $f(n)\to \infty$ possibly very slowly. Is there a graph of infinite chromatic number such that every finite subgraph on $n$ vertices can be made bipartite by deleting at most $f(n)$ edges?

Is $\sum_{n=1}^\infty \frac{p_n}{2^n}$ irrational? Here $p_n$ is the $n$-th prime ($p_1=2, p_2=3, \dots$).