Ramsey properties of algebraic graphs and hypergraphs

TL;DR

This paper investigates the limitations of algebraic graphs and hypergraphs with good Ramsey properties, establishing that such graphs require large parameters, and introduces a polynomial regularity lemma for algebraic hypergraphs.

Contribution

The paper extends existing results by proving lower bounds on parameters for algebraic graphs and hypergraphs with strong Ramsey properties, and develops a polynomial regularity lemma for algebraic hypergraphs.

Findings

Algebraic graphs with good Ramsey properties must have large complexity parameters.

Established lower bounds for clique and independent set sizes in algebraic graphs and hypergraphs.

Introduced a polynomial regularity lemma for algebraic hypergraphs defined by a single polynomial.

Abstract

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.