A bound for s-distance permutation families and explicit Ramsey graphs

Cameron gave an upper bound for the size of any s-distance family of permutations. We prove a modulo p version of Cameron’s result. In the proof we use the polynomial subspace method. As an application we describe here an explicit construction which produces for every integer m > 1 a graph on at least m 2 9 logm log logm vertices containing neither a clique of size m nor an independent set of size m.