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All Random Solved Random Open
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 \[f(N) \leq \log_2 N+o(\log\log N)?\]
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.