  We show that, assuming the (deterministic) Exponential Time Hypothesis,
distinguishing between a graph with an induced $k$-clique and a graph in which
all k-subgraphs have density at most $1-\epsilon$, requires $n^{\tilde
\Omega(log n)}$ time. Our result essentially matches the quasi-polynomial
algorithms of Feige and Seltser [FS97] and Barman [Bar15] for this problem, and
is the first one to rule out an additive PTAS for Densest $k$-Subgraph. We
further strengthen this result by showing that our lower bound continues to
hold when, in the soundness case, even subgraphs smaller by a near-polynomial
factor ($k' = k 2^{-\tilde \Omega (log n)}$) are assumed to be at most
($1-\epsilon$)-dense.
  Our reduction is inspired by recent applications of the "birthday repetition"
technique [AIM14,BKW15]. Our analysis relies on information theoretical
machinery and is similar in spirit to analyzing a parallel repetition of
two-prover games in which the provers may choose to answer some challenges
multiple times, while completely ignoring other challenges.
