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Ramsey-type problems for generalised Sidon sets

Christian Reiher, Vojtěch Rödl, Mathias Schacht

TL;DR

The paper constructs infinite $B_{k,\ell\l}$-sets in $\mathbb{N}$ that satisfy robust Ramsey-type properties and strong local-global regularity. It blends additive representation theory with ordered Ramsey theory by encoding $k$-term sums via differences along edges of a carefully chosen graph $G$ that contains monochromatic copies of generalised theta graphs $\Theta_{k,\ell}$. A base-$m$ arithmetic lemma guarantees uniqueness structures of sums, while a girth Ramsey theorem for ordered graphs provides the necessary Ramsey machinery; together these yield a set with both the $(k,\ell\l)$-Ramsey property and tight control over representations. The results extend classical constructions (Erdős–Newman, Nešetřil–Rödl) to general $k$ and $\ell\l$ and address a local-global question in Sidon-type sets, showing that local density does not compel a simple global decomposition. The constructed $X$ exhibits precise bounds on sub-Sidon structures and unique-sum behavior, with potential implications for additive combinatorics and Ramsey-type questions on integers.

Abstract

We establish the existence of generalised Sidon sets enjoying additional Ramsey-type properties, which are motivated by questions of Erdős and Newman and of Alon and Erdős.

Ramsey-type problems for generalised Sidon sets

TL;DR

The paper constructs infinite -sets in that satisfy robust Ramsey-type properties and strong local-global regularity. It blends additive representation theory with ordered Ramsey theory by encoding -term sums via differences along edges of a carefully chosen graph that contains monochromatic copies of generalised theta graphs . A base- arithmetic lemma guarantees uniqueness structures of sums, while a girth Ramsey theorem for ordered graphs provides the necessary Ramsey machinery; together these yield a set with both the -Ramsey property and tight control over representations. The results extend classical constructions (Erdős–Newman, Nešetřil–Rödl) to general and and address a local-global question in Sidon-type sets, showing that local density does not compel a simple global decomposition. The constructed exhibits precise bounds on sub-Sidon structures and unique-sum behavior, with potential implications for additive combinatorics and Ramsey-type questions on integers.

Abstract

We establish the existence of generalised Sidon sets enjoying additional Ramsey-type properties, which are motivated by questions of Erdős and Newman and of Alon and Erdős.
Paper Structure (6 sections, 3 theorems, 23 equations, 2 figures)

This paper contains 6 sections, 3 theorems, 23 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Ramsey-type property
  3. Local-global structure
  4. Main result
  5. Local Ramsey theory
  6. Proof of the main theorem

Key Result

Theorem 1.1

For all integers $k\geq 2$, $\ell\l\geq 2$, there exists an infinite $B_{k,\ell\l}$-set $X\subseteq \mathds N$ satisfying the following properties: In addition $X$ also satisfies:

Figures (2)

  • Figure 2.1: Ordered theta graph $\Theta_{5,3}$.
  • Figure 2.2: Some subforests fail to be forests.

Theorems & Definitions (9)

  • Theorem 1.1
  • Proposition 2.1
  • Definition 2.2: Forests of copies
  • Theorem 2.3: Girth Ramsey theorem
  • proof : Proof of Proposition \ref{['prop:21']}
  • proof
  • proof : Proof of Theorem \ref{['thm:B-set']}
  • Claim 3.2
  • proof