Ramsey properties of algebraic graphs and hypergraphs

One of the central questions in Ramsey theory asks how small the largest clique and independent set in a graph on N vertices can be. By the celebrated result of Erdas from 1947, a random graph on N vertices with edge probability 1/2 contains no clique or independent set larger than 2 log(2) N, with high probability. Finding explicit constructions of graphs with similar Ramsey-type properties is a famous open problem. A natural approach is to construct such graphs using algebraic tools. Say that an r-uniform hypergraph H is algebraic of complexity (n, d, m) if the vertices of H are elements of F-n for some field F, and there exist m polynomials f1, ..., f(m) : (F-n)(r) -> F of degree at most d such that the edges of H are determined by the zero-patterns of f(1), ..., f(m). The aim of this paper is to show that if an algebraic graph (or hypergraph) of complexity (n, d, m) has good Ramsey properties, then at least one of the parameters n, d, m must be large. In 2001, Ronyai, Babai and Ganapathy considered the bipartite variant of the Ramsey problem and proved that if G is an algebraic graph of complexity (n, d, m) on N vertices, then either G or its complement contains a complete balanced bipartite graph of size Omega(n,d,m)(N1/(n+1)). We extend this result by showing that such G contains either a clique or an independent set of size N-Omega(1/ndm) and prove similar results for algebraic hypergraphs of constant complexity. We also obtain a polynomial regularity lemma for r-uniform algebraic hypergraphs that are defined by a single polynomial that might be of independent interest. Our proofs combine algebraic, geometric and combinatorial tools.