Ramsey properties of algebraic graphs and hypergraphs.

One of the central questions in Ramsey theory asks how small can be the size of the largest clique and independent set in a graph on $N$ vertices. By the celebrated result of Erd\H{o}s from 1947, the 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 $\mathcal{H}$ is \emph{algebraic of complexity $(n,d,m)$} if the vertices of $\mathcal{H}$ are elements of $\mathbb{F}^{n}$ for some field $\mathbb{F}$, and there exist $m$ polynomials $f_1,\dots,f_m:(\mathbb{F}^{n})^{r}\rightarrow \mathbb{F}$ of degree at most $d$ such that the edges of $\mathcal{H}$ are determined by the zero-patterns of $f_1,\dots,f_m$. 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, R\'onyai, 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 $\Omega_{n,d,m}(N^{1/(n+1)})$. 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.