Additive combinatorics is a subfield of combinatorics, and so it is no surprise that graph theory plays an important role in this theory. Graph theory has already made an implicit appearance in previous chapters, most notably in the proof of the Balog–Szemerédi–Gowers theorem (Theorem 2.29). However there are several further ways in which graph theoretical tools can be utilized in additive combinatorics. We will only discuss a representative sample of these applications here. First we discuss Turán's theorem, which shows that sparse graphs contain large independent sets, and which is useful for constructing sum-free sets. Next we give a very brief tour of Ramsey theory, which allows one to find monochromatic structures in colored graphs (or other colored objects), in particular allowing one to find monochromatic progressions in any coloring of the integers (van der Waerden's theorem). Then we use some results about connectivity of dense graphs to establish the Balog–Szemerédi–Gowers theorem, which relates partial sum sets to complete sum sets and which has already been exploited in Chapter 2. Finally, we use the theory of commutative directed graphs to establish the Plünnecke inequalities, which are perhaps the sharpest inequalities known for sum sets and which strengthen several of the results already established in Chapter 2.
In Chapter 10 and Chapter 11 we shall discuss one final graph-theoretical tool, the Szemerédi regularity lemma, which has had many applications in several areas of discrete mathematics, but which in additive combinatorics has had an especially crucial role in the study of arithmetic progressions in dense sets.
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