AbstractAn algorithm for the construction of Ramsey graphs with a given automorphism groupGis presented. To find a graph onnvertices with no clique ofkvertices,Kk, and no independent set oflvertices, ¯Kl,k, l≤n, with an automorphism groupG, a Boolean formula α based on theG‐orbits ofk‐subsets andl‐subsets of vertices is constructed from incidence matrices belonging toG. This Boolean formula is satisfiable if and only if the desired graph exists, and each satisfying assignment to α specifies a set of orbits of pairs of vertices whose union gives the edges of such a graph. Finding these assignments is basically equivalent to the conversion of α from CNF to DNF (conjunctive to disjunctive normal form). Though the latter problem is NP‐hard, we present an “efficient” method to do the conversion for the formulas that appear in this particular problem. WhenGis taken to be the dihedral groupDnforn≤ 101, this method matches all of the previously known cyclic Ramsey graphs, as reported by F. R. K. Chung and C. M. Grinstead [“A Survey of Bounds for Classical Ramsey Numbers,”Journal of Graph Theory,7(1983), 25–38], in dramatically smaller computer time when compared to the time required by an exhaustive search. Five new lower bounds for the classical Ramsey numbers are established:R(4, 7) ⩾ 47,R(4, 8) ⩾ 52,R(4, 9) ⩾ 69,R(5,7) ⩾ 76, andR(5, 8) ⩾ 94. Also, some previously known cyclic graphs are s