Szemerédi's regularity lemma application on 3-term arithmetic progression

In this paper, the Szemeredi's Regularity Lemma and its application are studied. This lemma is used to partition a large enough graph into almost equal parts so that the number of edges across the parts is fairly random. On the other hand, Roth's Theorem states that there exists an arithmetic progression with length 3 in a subset in integer with positive upper density. We shall see that it can be proved by using triangle removal lemma, which is an application of Szemeredi's Regularity Lemma. The regularity lemma does not seem to be a direct tool to be used on Roth's theorem. The lemma deals with the graphs while Roth's theorem states about subsets on integers. But however, the two are connected through the construction of an auxiliary graph, where 3-arithmetic progression of integers subsets corresponds to triangles in this such graph. At the end of this explanation, we will find that the whole trivial triangles formed by such graph partition are all disjoint. This is at the point we can conclude that there exists an arithmetic progression with length 3 in a subset in integer with positive upper density.