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.
Categories: math.CO math.NT