On a Lemma of Scarf

Abstract

The aim of this note is to point out some combinatorial applications of a lemma of Scarf, proved first in the context of game theory. The usefulness of the lemma in combinatorics has already been demonstrated in [1], where it was used to prove the existence of fractional kernels in digraphs not containing cyclic triangles. We indicate some links of the lemma to other combinatorial results, both in terms of its statement (being a relative of the Gale-Shapley theorem) and its proof (in which respect it is a kin of Sperner’s lemma). We use the lemma to prove a fractional version of the Gale-Shapley theorem for hypergraphs, which in turn directly implies an extension of this theorem to general (not necessarily bipartite) graphs due to Tan [12]. We also prove the following result, related to a theorem of Sands, Sauer and Woodrow [10]: given a family of partial orders on the same ground set, there exists a system of weights on the vertices, which is (fractionally) independent in all orders, and each vertex is dominated by them in one of the orders.

Research supported by the fund for the promotion of research at the Technion and by a grant from the Israel Science Foundation.

Part of the research was done at the Centrum voor Wiskunde en Informatica (CWI) and during two visits to the Technion, Haifa. Research was supported by the Netherlands Organization for Scientific Research (NWO) and by OTKA T 029772.