The k-Satisfiability problem remains NP-complete for dense families

We consider the ~ATISFIABILITY problem (~-SAT): Given a family F of n clauses cl, ._, , c, in conjunctive normal form, each consisting of k literals corresponding to k different variables of a set of r 2 k 2 1 boolean variables, is F satisfiable? By k-SAT@ no) we denote the k-SAT problem restricted to families with n > n,(r) clauses. We prove that for each k > 3 and each integer I > 4 such that r > Ik2, the k-SAT(>(;) (2’l-4/1)) problem is NP-complete.