Hardness of Finding Independent Sets in Almost q-Colorable Graphs

We show that for any ε >; 0, and positive integers k and q such that q ≥ 2^k + 1, given a graph on N vertices that has a q-colorable induced subgraph of (1 - ε)N vertices, it is NP-hard to find an independent set of N/q^k+1 vertices. This substantially improves upon the work of Dinur et al. [1] who gave a corresponding bound of N/q². Our result implies that for any positive integer k, given a graph that has an independent set of ≈ (2^k + 1)^-1 fraction of vertices, it is NP-hard to find an independent set of (2^k + 1)^-(k+1) fraction of vertices. This improves on the previous work of Engebretsen and Holmerin [2] who proved a gap of ≈ 2^-k vs 2^-(k:2), which is best possible using techniques (including those of [2]) based on the query efficient PCP of Samorodnitsky and Trevisan [3].