Roth-type theorems in additive combinatroics

TL;DR

This paper explores Roth-type theorems in additive combinatorics, discussing their origins from van der Waerden's theorem and Erdős-Turán conjecture, and providing concise expositions of proof ideas.

Contribution

It reviews and summarizes various Roth-type theorems, highlighting their development and proof strategies in additive combinatorics.

Findings

List of Roth-type theorems and their relationships

Insights into proof techniques for Roth-type theorems

Historical overview of developments in additive combinatorics

Abstract

In this article we will introduce a central problem in additive combinatorics, which arised from the famous van der Waerden theorem and an early conjecture of Erd\H{o}s and Tur\'{a}n. The first important theorem was due to Roth in 1953. There were a number of generalized or improved results afterwards, which we call Roth-type theorems. We will list them and try to give concise expositions to the ideas in some of the proofs without much prior knowledge.