On a lemma of Scarf

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 a paper by the first author and R. Holzman (J. Combin. Theory Ser. B 73 (1) (1998) 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 Spernerʹs 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 (J. Algorithms 12 (1) (1991) 154). We also prove the following result, related to a theorem of Sands et al. (J. Combin. Theory Ser. B 33 (3) (1982) 271): 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.