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

Abstract

A very nice result of Bárány and Lehel asserts that every finite subset X or R d 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.

Three-chromatic geometric hypergraphs

“…That proof, however, only worked when C was a disk, and while the generalization to other convex bodies with a smooth boundary seemed feasible, we saw no way to extend it to arbitrary convex bodies. The proof of Theorem 1 relies on a surprising connection to two other famous results, the solution of the two dimensional case of the Illumination conjecture [22], and a recent solution of the Erdős-Sands-Sauer-Woodrow conjecture by Bousquet, Lochet and Thomassé [7]. In fact, we need a generalization of the latter result, which we prove with the addition of one more trick to their method; this can be of independent interest.…”

“…Theorem 9 (Bousquet, Lochet, Thomassé [7]). For every k, there exists an integer f (k) such that if D is a complete multidigraph whose arcs are the union of k quasi orders, then D has a dominating set of size at most f (k).…”

“…Let us quickly recap the main steps from the proof of Theorem 9 from [7]. For a function w : V → R and S ⊂ V let w(S) = x∈S f (x).…”

A lower bound on the size of an absorbing set in an arc-coloured tournament

“…Theorem 1 (Bousquet, Lochet and Thomassé [2]). Function f is well defined and f (n) = O(ln(n) · n n+2 ).…”

“…Recently, Pálvőlgyi and Gyárfás [6] have shown that a positive answer to this problem would imply a new proof of a former result from Bárány and Lehel [1] stating that any set X of points in R d can be covered by f (d) X-boxes (each box is defined by two points in X). In 2017, Bousquet Lochet and Thomassé [2] gave a positive answer to the problem of Sands, Sauer and Woodrow.…”