Ramsey theory of low-degree semialgebraic relations

TL;DR

This paper establishes that hypergraphs defined by low-degree polynomial inequalities inherently contain large homogeneous subsets, linking algebraic complexity with combinatorial structure.

Contribution

It proves a Ramsey-type theorem for hypergraphs defined by low-degree semialgebraic relations, extending classical combinatorial results to algebraic settings.

Findings

Hypergraphs with low-degree polynomial inequalities have large homogeneous subsets.

The size of the largest homogeneous set is bounded by a tower function depending on the polynomial degree.

The result generalizes classical Ramsey theory to algebraic hypergraph structures.

Abstract

We prove that hypergraphs defined by low-degree polynomial inequalities contain large homogeneous subsets. Formally, let $H$ be an $r$-uniform hypergraph on $N$ vertices that is semialgebraic of constant description complexity, and each defining polynomial has degree at most $D$. Then $H$ contains a clique or an independent set of size $n$, where $N\leq \mbox{tw}_{3D^3}(n)$.