A graph G = (V,E) is said to be (k,δ) almost colorable if there is a subset of vertices V' ⊆ V of size at least (1-δ)|V| such that the induced subgraph of G on V' is k-colorable. We prove that for all k, there exists δ > 0 such for all ε > 0, given a graph G it is NP-hard (under randomized reductions) to distinguish between:

1) Yes case: G is (3,ε) almost colorable.

2) No case: G is not (k,δ) almost colorable. This improves upon an earlier result of Khot et al. [Irit Dinur et al., 2018], who showed a weaker result wherein in the "yes case" the graph is (4,ε) almost colorable.