Let $h(n)$ be minimal such that any group $G$ with the property that any subset of $>n$ elements contains some $x\neq y$ such that $xy=yx$ can be covered by at most $h(n)$ many Abelian subgroups.
Pyber [Py87] has proved there exist constants $c_2>c_1>1$ such that $c_1^n<h(n)<c_2^n$. Erdős [Er97f] writes that the lower bound was already known to Isaacs.
OPEN
If $G$ is an abelian group then can there exist an exact covering of $G$ by more than one cosets of different sizes? (i.e. each element is contained in exactly one of the cosets)
A problem of Herzog and Schönheim. In [Er97c] Erdős asks this for finite (not necessarily abelian) groups.
Additional thanks to: Boris Alexeev and Dustin Mixon
OPEN
Define $f(N)$ be the minimal $k$ such that the following holds: if $G$ is an abelian group of size $N$ and $A\subseteq G$ is a random set of size $k$ then, with probability $\geq 1/2$, all elements of $G$ can be written as $\sum_{x\in S}x$ for some $S\subseteq A$. Is
\[f(N) \leq \log_2 N+o(\log\log N)?\]
