Flag algebras: An interim report

Abstract

For the most part, this article is a survey of concrete results in extremal combinatorics obtained with the method of flag algebras. But our survey is also preceded, interleaved and concluded with a few general digressions about the method itself. Also, instead of giving a plain and unannotated list of results, we try to divide our account into several connected stories that often include historical background, motivations and results obtained with the help of methods other than flag algebras.