Colourful categories

This paper presents Ramsey theory in category-theoretic terms as a message from a non-expert author to a non-expert reader. Everything is explained starting from the level zero, and an attempt is made to be as self-explanatory in the terminology and notation as possible. For the sake of references, the paper also reproduces traditional terminology, with concepts and theorems often named after (presumed) discoverers who are largely unknown to outsiders to the field. The sources are referred to in a manner so as to make them easy to find on the web; only exceptionally are non-freely accessible items referred to. Certain questions are formulated as ‘conjectures’, not out of a deep belief in their validity but because they sound better stated explicitly. There are no new results in this article, no deepening of particular aspects of Ramsey theory, no attempts to be comprehensive. But, in the spirit of the ideas of Anatoly Vershik, an attempt is made to move transversally across common directions of research, to see interrelations between them and to formulate questions. In fact, the article reproduces a chapter from the author's as yet unfinished manuscript “A number of questions”.\footnote{http://www.ihes.fr/~gromov/PDF/Problems-marc6-11-2014.pdf.}
Bibliography: 65 titles.