Hardness of approximating hitting set
For every choice of $\alpha &gt; 0$, Moshkovitz shows that approximating set cover on instances with $n^{C_\alpha}$, where $C_\alpha$ is some constant depending only on $\alpha$, better than $(1-\alpha) \log n$ is NP-hard. Since $\log m \leq C_\alpha \log n$, it is NP-hard to approximate set cover better than $\frac{1-\alpha}{C_\alpha} \log m$. I'm not sure what is $\sup_{\alpha\to 1} \frac{1-\alpha}{C_\alpha}$, but it can probably be gleaned from the paper. At any rate, all three results you mention are correct, and the first can be strengthened by replacing $c$ with $1-\alpha$ for any $\alpha &gt; 0$.