A proof of the Erdős-Sands-Sauer-Woodrow conjecture

A very nice result of Bárány and Lehel asserts that every finite subset X or R can be covered by f(d) X-boxes (i.e. each box has two antipodal points in X). As shown by Gyárfás and Pálvőlgyi this result would follow from the following conjecture : If a tournament admits a partition of its arc set into k quasi orders, then its domination number is bounded in terms of k. This question is in turn implied by the Erdős-Sands-Sauer-Woodrow conjecture : If the arcs of a tournament T are colored with k colors, there is a set X of at most g(k) vertices such that for every vertex v of T , there is a monochromatic path from X to v. We give a short proof of this statement. We moreover show that the general Sands-Sauer-Woodrow conjecture (which as a special case implies the stable marriage theorem) is valid for directed graphs with bounded stability number. This conjecture remains however open.