All Random Solved Random Open
0 solved out of 3 shown
Let $G$ be a group and let $\Gamma(G)$ be the commuting graph of $G$, with vertex set $G$ and $x\sim y$ if and only if $xy=yx$. Let $h(n)$ be minimal such that if the largest independent set of $\Gamma(G)$ has at most $n$ elements then $\Gamma(G)$ can be covered by $h(n)$ many complete subgraphs. Estimate $h(n)$ as well as possible.
Pyber [Py87] has proved there exist constants $c_2>c_1>1$ such that $c_1^n<h(n)<c_2^n$.
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.
Additional thanks to: Boris Alexeev and Dustin Mixon
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 \[f(N) \leq \log_2 N+o(\log\log N)?\]
Erdős and Rényi [ErRe65] proved that \[f(N) \leq \log_2N+O(\log\log N).\] Erdős believed improving this to $o(\log\log N)$ is impossible.