Near-perfect clique-factors in sparse pseudorandom graphs

We prove that, for any, there exists a constant c = c(t) > 0 such that any d-regular n-vertex graph with the second largest eigenvalue in absolute value λ satisfying contains vertex-disjoint copies of kt covering all but at most vertices. This provides further support for the conjecture of Krivelevich, Sudakov and Szábo (Combinatorica 24 (2004), pp. 403-426) that (n, d, λ)-graphs with n ∈ 3ℕ and for a suitably small absolute constant c > 0 contain triangle-factors. Our arguments combine tools from linear programming with probabilistic techniques, and apply them in a certain weighted setting. We expect this method will be applicable to other problems in the field.