Removal lemmas in sparse graphs

In this work we explain and prove the graph removal lemma, both in its dense and sparse cases, and show how these can be applied to finite groups to obtain arithmetic removal lemmas. We show how the concept of regularity plays a crucial role in the proof of the removal lemma. We explain the motivation behind the sparse case, and the importance of pseudorandom graphs in sparse versions of the removal lemma. Finally, we show how the removal lemma, both in its graph and arithmetic versions, can be used to prove Roth's theorem, that is, the existence of 3-term arithmetic progressions in any dense subset of the natural numbers