An $\tilde Ω(\log n \log m)$ Information-Theoretic Lower Bound for Randomized Online Set Cover

We show an information-theoretic lower bound of $Ω\left(\frac{\log n \log m}{\log \log n + \log \log m}\right)$ for online set cover against randomized algorithms, for all sufficiently large $m$ and $n$ satisfying $\log^2 n \leq m \leq 2^n$.