Transversal Theory

1. Introduction. Transversal theory is a branch of combinatorial mathematics which is only just beginning to emerge as a reasonably connected and coherent subject. Whether this is yet rich enough or mature enough to be called a 'theory' may be a matter for debate; indeed, it is by no means certain that this part of mathematics may not finally be classified under some broader, more comprehensive title. However, what is beyond dispute is the fact that during the last two decades a large number of papers have been published which include some reference to the so-called marriage theorem (Theorem 2.1), which is the starting point for transversal theory. These papers deal with surprisingly diverse problems and their only connecting link seems to be this common reference to the marriage theorem. The arguments employed have generally had an ad hoc flavour although some of these have been highly original. Transversal theory is a depository for developing those mathematical ideas of the marriage theorem type which frequently recur and which seem to belong to some more general framework. Two books on the subject have been published recently by Crapo and Rota [11] and Mirsky [44] although these were written from rather differing viewpoints. The first part of this article will be expository and cover ground which is familiar to most combinatorial mathematicians. In the second part I shall describe some more recent work done on infinite transversals. The earlier bibliography, detailed proofs and a historical commentary can be found in Mirsky's book. Apart from the new result in set theory mentioned in § 6,1 shall not dwell upon the applications of transversal theory to other branches of mathematics, but refer the reader interested in this aspect to the article by Harper and Rota [31]. Instead I shall try to give emphasis to those results which are either new or which have influenced the development of the subject.