Ramsey graphs cannot be defined by real polynomials

Let P(x,y,n) be a real polynomial and let {G_n} be a family of graphs, where the set of vertices of G_n is {1,2,…,n} and for 1 ≤ i < j ≤ n {i,j} is an edge of G_n iff P(i,j,n) > 0. Motivated by a question of Babai, we show that there is a positive constant c depending only on P such that either G_n or its complement G_n contains a complete subgraph on at least c2^{1/2√log n} vertices. Similarly, either G_n or G_n contains a complete bipartite subgraph with at least cn^1/2 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.