In this paper we put forward a conjecture: an instantiation of the Sliding Scale Conjecture of Bellare, Goldwasser, Lund and Russell to projection games. We refer to this conjecture as the Projection Games Conjecture.

We further suggest the research agenda of establishing new hardness of approximation results based on the conjecture. We pursue this line of research by establishing a tight NP-hardness result for the Set-Cover problem. Specifically, we show that under the projection games conjecture (in fact, under a quantitative version of the conjecture that is only slightly beyond the reach of current techniques), it is NP-hard to approximate Set-Cover on instances of size N to within $(1-x)ln N$ for arbitrarily small $x>0$. We do this by modifying Feige's reduction that gives a $(1-x)ln N$ inapproximability under the assumption that $NP$ is not contained in $DTIME(N^{O(\log\log N)})$.