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].

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 $H(n)-\log_2 n \to \infty$.

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

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

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 $g(n) \geq (1-o(1))n$?

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 $\mathcal{F}$ 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 $\lvert \mathcal{F}'\rvert \leq \lvert \{ A\in \mathcal{F} : x\in A\}\rvert.$

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

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

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 there a graph of infinite chromatic number such that every finite subgraph on $n$ vertices can be made bipartite by deleting at most $\sqrt{n}$ edges?

Let $\mathfrak{m}$ be an infinite cardinal and $G$ be a graph with chromatic number $\mathfrak{m}$. Let $r\geq 1$. Must $G$ contain a subgraph of chromatic number $\mathfrak{m}$ which does not contain any odd cycle of length $\leq r$?

Let $A\subseteq \mathbb{N}$ be such that $A+A$ has positive lower density. Can one always decompose $A=A_1\sqcup A_2$ such that $A_1+A_1$ and $A_2+A_2$ both have positive lower density?

Murty-Simon Conjecture*

Let $G$ be a graph on $n$ vertices with diameter $2$ such that deleting any edge increases the diameter. Is it true that $G$ has at most $\lfloor n^2 / 4 \rfloor$ edges? Equality is conjectured to hold for the complete balanced bipartite graph $K_{\lceil n/2 \rceil, \lfloor n/2 \rfloor}$.

The conjecture is resolved up to a finite check: Fan [Fa87] verified it for $n \leq 24$ and $n = 26$, and Füredi [Fü92] proved it for all sufficiently large $n$.

Is there a graph of chromatic number ℵ_ 1 with ℵ_ 1 vertices such that for all ε > 0, if n is sufficiently large and H is a subgraph on n vertices, then H contains an independent set of size > n ^ (1 - ε)?

What is the supremum of the set of admissible numbers?

Is every proportionately dissociated (infinite) set the union of a finite number of dissociated sets?

Let $h(n)$ be maximal such that if $A\subseteq \mathbb{Z}$ with $\lvert A\rvert=n$ then there is $B\subseteq A$ with $\lvert B\rvert \geq h(n)$ such that if $a_1+\cdots+a_r=b_1+\cdots+b_s$ with $a_i,b_i\in B$ then $r=s$.

Estimate $h(n)$.

Let $h(n)$ be maximal such that if $A\subseteq \mathbb{Z}$ with $\lvert A\rvert=n$ then there is $B\subseteq A$ with $\lvert B\rvert \geq h(n)$ such that if $a_1+\cdots+a_r=b_1+\cdots+b_s$ with $a_i,b_i\in B$ then $r=s$.

Is $h(n) = \Theta(\sqrt{n})$?

By the solved variant erdos_789.variants.isBigO_sq, in order to prove erdos_789.variants.sq it suffices to show $\sqrt{n}=O(h(n))$.