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 Erdős 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
$\mathcal {H}$
is 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, Rónyai, 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.