Ramsey Graphs Cannot

Let P(x,y,n) be a real polynomial and let {G, } be a family of graphs, where the set of vertices of G, is (1.2,.. . ,n} and for 1 I i < j' 5 n {;,I] is an edge of G, iff P(i,j,n) > 0. Motivated by a question of Babai, we show that there is a positiye constant c depending only on P such that either G or its complement G, contains _a complete subgraph on at least c2 vertices. Similarly, either G, or G, contains a complete bipartite subgraph with at least cntn vertices in each color class. Similar results are proved for graphs defined by real polynomials in a more general way, showing that such graphs satisfy much stronger Ramsey bounds than do random graphs. This may partially explain the difficulties in finding an explicit construction for good Ramsey graphs.