  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.
