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 − , requires n ( logn ) 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 − ( logn ) ) are assumed to be at most ( 1 − )-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.