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… CONTINUE READING