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Combinatorial formulae for multiple set-valued labellings. (English) Zbl 0791.05007
We develope the geometric notions of general position maps, $$\pi$$- balanced and $$\pi$$-subbalanced sets and then apply them to prove two general combinatorial formulae for multiple set-valued labellings on simplices related to the celebrated Sperner combinatorial lemma [Abh. Math. Semin. Univ. Hamb. 6, 265-272 (1928; JFM 54.0614.01)]. We apply one of the combinatorial formulae to covering theory of simplices and obtain a new covering theorem which is a common generalization of the Shapley theorem and the Gale theorem.

##### MSC:
 05A99 Enumerative combinatorics 52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.) 57Q65 General position and transversality 05D05 Extremal set theory
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##### References:
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