A compactness argument in the additive theory and the polynomial method

In this expository paper we collect some combinatorial problems in the additive theory that can be easily solved in ordered Abelian groups. We study how such results, obtained by simple combinatorial arguments, can be extended to other Abelian groups. In many cases, best results can be obtained with the help of the so-called polynomial method that has evolved to a very powerful tool in the additive theory during the last decade.