Lower bounds for Ramsey numbers of KnKn with a small subgraph removed

Let FF be a graph of order at most nn, size at most n−2n−2 and no isolated vertex, and Kn−FKn−F a graph obtained from KnKn by deleting edges of a copy of FF. Denote QpQp the Paley graph of order pp. It is shown that if QpQp contains no Kn−FKn−F, then the Ramsey number r(Kn+1−F)≥2p+1r(Kn+1−F)≥2p+1.