Open-problem statements, with their sources attached.

For any tree $T$ with $n$ edges, the complete graph $K_{2n+1}$ decomposes into $2n+1$ edge-disjoint copies of $T$ via cyclic shifts of a single embedding.

The $2n+1$ copies are $f_0, f_1, \dots, f_{2n}$ where $f_i(v) = f_0(v) + i$ for all vertices $v$ , each copy is obtained by adding $i \pmod{2n+1}$ to every vertex of the base copy. This is strictly stronger than RingelConjecture.ringel_conjecture.

Is it true that, for all $k\neq 1$, there are infinitely many $n$ such that $2^n\equiv k\pmod{n}$?

Conjecture 3.2 in [Wa2011]: Each Latin square of odd order has at least one transversal.

For each $n \in \mathbb{N}$ choose some $X_n \subseteq \mathbb{Z}/n\mathbb{Z}$. Let $B = \{m \in \mathbb{N} : \forall n, m \not\equiv x \pmod{n} \text{ for all } x \in X_n\}$. Must $B$ have a logarithmic density?

The smallest odd number for which this conjecture is not known is 11.

Let $A\subseteq \mathbb{N}$ be a set such that $\lvert A\cap [1,x]\rvert=o(x^{1/2})$. Let $B=\{ n\geq 1 : a\nmid n\textrm{ for all }a\in A\}$. If $B=\{b_1 < b_2 < \cdots\}$ then is it true that $\lim_{x \to \infty} \frac{1}{x}\sum_{b_i < x}(b_{i+1}-b_i)^2$ exists (and is finite)?

For example, when $A=\{p^2: p\textrm{ prime}\}$ then $B$ is the set of squarefree numbers, and the existence of this limit was proved by Erdős.

See also [208].

Conjecture 5.1 in [Wa2011]: Every latin square has a near-transversal

Let $\alpha,\beta \in \mathbb{R}$. Is it true that$\liminf_{n\to \infty} n \| n\alpha \| \| n\beta\| =0$? This is also known as the Littlewood conjecture.

Conjecture 6.7 in [Wa2011]: There exist real constants $0 < c_1 < c_2 < 1$ such that

for all odd $n \geq 3$.

Let $f$ be the asymptotic distribution function of $\varphi(n)/n$, so that for each $c \in [0,1]$, $f(c)$ is the natural density of $\{n : \varphi(n) < cn\}$. Is it true that there is no $x$ such that the derivative $f'(x)$ exists and is positive?

Conjecture 6.9 in [Wa2011]:

It is not even known if this limit exists. Note that $z_n = 0$ for even $n$ (see z_even), so the limit must be restricted to odd $n$; here we parametrise odd $n$ as $2k + 1$.

Is there an infinite set $A \subset \mathbb{N}$ such that for every $a \in A$, there is an integer n such that $\phi(n)=a$, and yet if $n_a$ is the smallest such integer, then $\frac{n_a}{a} → \infty$ as $a → ∞$?

MOLS existence problem: determine exactly which orders n admit a complete set of n - 1 mutually orthogonal latin squares.

Equivalently, this asks for which orders affine planes of order n exist. Complete sets are known for prime-power orders; the smallest currently unresolved order is 12.

Chowla's cosine problem*

If $A\subset \mathbb{N}$ is a finite set of positive integers of size $N > 0$ then is there some absolute constant $c>0$ and $\theta$ such that $\sum_{n\in A}\cos(n\theta) < -cN^{1/2}?$

The smallest unresolved case of the MOLS existence problem: whether there are 11 mutually orthogonal latin squares of order 12.

Let $A$ be a finite set of integers. Is it true that for every $\epsilon>0$ $\max( \lvert A+A\rvert,\lvert AA\rvert)\gg_\epsilon \lvert A\rvert^{2-\epsilon}?$

The Latin Tableau Conjecture: If G is the simple graph of a Young diagram, then G is CDS-colorable.

Let $f$ be a Rademacher multiplicative function. Does there exist some constant $c > 0$ such that, almost surely,

For $N = 6$ and $D = 3$, does there exist no solution to the monochromatic quantum graph equation system over $\mathbb{C}$?

Let $\epsilon>0$ and $N$ be sufficiently large. Is it true that if $A\subseteq \{1,\ldots,N\}$ has size at least $\epsilon N$ then there must be distinct $a,b,c\in A$ such that $[a, b]=[b, c]=[a, c],$ where $[\cdot, \cdot]$ denotes the least common multiple?