On a Lemma of Scarf

  • Ron Aharoni
  • Tamás Fleiner
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2337)


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.


Partial Order Stable Match Maximum Cardinality Hasse Diagram Cycle Component 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Ron Aharoni
    • 1
  • Tamás Fleiner
    • 2
    • 3
  1. 1.TechnionHaifa
  2. 2.Alfréd Rényi Institute of MathematicsHungarian Academy of SciencesHungary
  3. 3.Operations Research DepartmentEötvös Loránd UniversityHungary

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