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Ramsey theory of low-degree semialgebraic relations

Azem Adibelli, István Tomon

TL;DR

The work addresses Ramsey-type questions for hypergraphs defined by low-degree semialgebraic relations. It develops a framework based on $D$-dependent decompositions, hypercube evaluations, and exponential separation to extract large homogeneous subsets, culminating in a tower-type bound whose height depends only on the polynomial degree $D$. The main result shows that $R^{d,D,m}_r(n)$ is bounded by a tower of height proportional to $D^3$, significantly sharpening general Ramsey bounds for this natural geometric class. The methods unify combinatorial Ramsey theory with semialgebraic geometry and yield implications for geometric combinatorics, where structured edge relations arise from polynomial inequalities.

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 $N\leq \mbox{tw}_{3D^3}(n)$.

Ramsey theory of low-degree semialgebraic relations

TL;DR

The work addresses Ramsey-type questions for hypergraphs defined by low-degree semialgebraic relations. It develops a framework based on -dependent decompositions, hypercube evaluations, and exponential separation to extract large homogeneous subsets, culminating in a tower-type bound whose height depends only on the polynomial degree . The main result shows that is bounded by a tower of height proportional to , significantly sharpening general Ramsey bounds for this natural geometric class. The methods unify combinatorial Ramsey theory with semialgebraic geometry and yield implications for geometric combinatorics, where structured edge relations arise from polynomial inequalities.

Abstract

We prove that hypergraphs defined by low-degree polynomial inequalities contain large homogeneous subsets. Formally, let be an -uniform hypergraph on vertices that is semialgebraic of constant description complexity, and each defining polynomial has degree at most . Then contains a clique or an independent set of size , where .
Paper Structure (11 sections, 10 theorems, 84 equations)

This paper contains 11 sections, 10 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Semialgebraic hypergraphs
  4. Exponential sequences
  5. Extremal combinatorics
  6. Proof outline
  7. Warm-up: degree 2 semialgebraic hypergraphs
  8. Low-degree semialgebraic hypergraphs
  9. Functions defined on tuples
  10. Exponentially separated functions on hypergraphs
  11. Proof of the Main theorem

Key Result

Theorem 1.1

Let $H$ be an $r$-uniform semialgebraic hypergraph of description complexity $(d,D,m)$ on $N$ vertices, where $N\gg d,D,m,r$. Then $H$ contains a clique or an independent set of size $n$, where

Theorems & Definitions (29)

  • Theorem 1.1
  • Definition 1: Semialgebraic hypergraph
  • Definition 2: Exponential sequence
  • Definition 3
  • Lemma 1
  • proof
  • Definition 4
  • Theorem 2.1
  • Lemma 2
  • proof
  • ...and 19 more