Combinatorics 3 - Combinatorial Number Theory

“Combinatorial number theory”, in very loose terms, can be described as an area of mathematics which is a cross between combinatorics and number theory. More precisely, the area concerns structures of integers (or similar sets), with some number theoretic properties, which can be studied mainly by combinatorial means, rather than, for example, by algebraic methods. This area really only truly emerged throughout the 20th century, but has become extremely popular in the last four decades or so. In recent years, more and more mathematical olympiad style problems related to the area have also appeared. In this set of notes, we shall consider many of the most well-known theorems in combinatorial number theory, and show some applications along the way. These notes can be considered as a continuation of the set titled “Combinatorics”, and any undefined graph theoretic terms here can be found in the preceding notes.