Chapter 2. Matching Theory. 2.1. What Is Matching Theory?

An elementary example of matching theory is the following puzzle (popularized by R. Gomory). Given a standard 8× 8 chessboard, can it always be covered by dominoes (a piece consisting of two squares) if one arbitrary black square and one arbitrary white square are deleted? The solution is given in the exercises. Another puzzle which involves the idea of matchings is the following Putnam problem from 1979. Let n red points r1, r2, . . . , rn and n blue points b1, b2, . . . , bn be given in the Euclidean plane. Show that there exists a permutation π of {1, 2, . . . , n}, matching the red points with the blue points, so that no pair of finite line segments ri bπ(i) and rj bπ(j) intersect. Although this puzzle is about matchings, it is perhaps not matching theory in our sense. The reason – a hint for its solution – is that it is really a geometry problem. 1 Our third example is indisputably a theorem in matching theory. 2 This theorem settles the question, often asked when one is first told that left and right cosets of a subgroup may not be the same, whether there exists a set of elements acting as simultaneous left and right coset representatives.