Clarification on the inapproximability of set cover
The hardness proof appears in an earlier paper of Moshkovitz, The Projection Games Conjecture and The NP-Hardness of ln n-Approximating Set-Cover; Dinur and Steurer proved a version of the Projection Games Conjecture, thus completing the NP-hardness proof. Moshkovitz clearly states that $n$ is the size of the universe, and she also assumes that the number of sets is polynomial in $n$.

Feige had proved earlier that if you can approximate set cover better than $\ln n$ then $\mathsf{NP}\subseteq \mathsf{DTIME}(N^{\log\log N})$. He improved on earlier result of Lund and Yannakakis which showed a hardness of $0.72\ln n$ under the same assumption.

The threshold $\ln n$ (where $n$ is the size of the universe) is important since it is achieved by the greedy algorithm. The analysis of the greedy algorithm shows that after $OPT \cdot t$ steps (where $OPT$ is the size of an optimal solution), we have covered a $1-e^{-t}$ fraction of the universe. When $e^{-t} &lt; 1/n$, we have covered the entire universe (since we cover more than $n-1$ elements). Hence after at most $t = OPT \cdot \ln n$ steps, the greedy algorithm manages to cover the entire universe. This is where the dependence on the size of the universe comes from.