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On Rado's single equation theorem

Tom Sanders

Abstract

We show that for non-zero integers $a$ and $b$ there is a natural number $N < \exp(r^{2+o_{a,b;r\rightarrow \infty}(1)})$ such that in any $r$-colouring of $\{1,\dots,N\}$ there are $x,y,z$, all in the same colour class, such that $ax-ay=bz$.

On Rado's single equation theorem

Abstract

We show that for non-zero integers and there is a natural number such that in any -colouring of there are , all in the same colour class, such that .
Paper Structure (11 sections, 16 theorems, 102 equations)

This paper contains 11 sections, 16 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. When can we do better?
  3. The toy
  4. Proof of Theorem \ref{['thm.mn']}
  5. Notation
  6. The main iteration lemma
  7. Bohr sets
  8. A local version of spectral positivity
  9. A local version of sifting
  10. A local version of Chang's theorem
  11. A local version of the Croot-Sisask almost periodicity lemma

Key Result

Theorem 1.1

Suppose that $a \in \mathbb{Z}^d$ is partition regular. Then

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 2.1
  • Proposition 2.2
  • proof : Proof of Theorem \ref{['thm.model']}
  • Theorem 3.1
  • proof : Proof of Theorem \ref{['thm.mn']} assuming Theorem \ref{['thm.keycount']}
  • Lemma 3.4: Iteration lemma
  • proof
  • Proposition 3.5
  • proof
  • ...and 20 more