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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.

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
Full Text: DOI EuDML
[1] Bapat, R.B.: A constructive proof of a permutation-based generalization of Sperner’s lemma. Math. Program.44, 113-120 (1989) · Zbl 0673.55004
[2] Brown, A.B., Cairns, S.S.: Strengthening of Sperner’s lemma applied to homology. Proc. Natl. Acad. Sci. USA47, 113-114 (1961) · Zbl 0097.38702
[3] Cohen, D.I.A.: On the Sperner lemma. J. Comb. Theory2, 585-587 (1967) · Zbl 0163.18104
[4] Fan, K.: A generalization of Tychonoff’s fixed point theorem. Math. Ann.142, 305-310 (1961) · Zbl 0093.36701
[5] Fan, K.: Simplicial maps from an orientablen-pseudomanifold intoS m with the octahedral triangulation. J. Comb. Theory2, 588-602 (1967) · Zbl 0149.41302
[6] Fan, K.: Some properties of convex sets related to fixed point theorems. Math. Ann.266, 519-537 (1984) · Zbl 0532.47043
[7] Gale, D.: Equilibrium in a discrete exchange economy with money. Int. J. Game Theory13, 61-64 (1984) · Zbl 0531.90011
[8] Gr?nbaurn, B.: Convex polytopes. London New York Sydney: Wiley 1967 · Zbl 0163.16603
[9] Hall, P.: On the representatives of subsets. J. Lond. Math. Soc.10, 26-30 (1935) · Zbl 0010.34503
[10] Knaster, B., Kuratowski, C., Mazurkiewicz, S.: Ein Beweis des Fixpunktsatzes f?rn-dimensionale Simplexe. Fundam. Math.14, 132-137 (1929) · JFM 55.0972.01
[11] Kuratowski, C.: A half century of Polish Mathematics. Warszawa: PWN-Polish Scientific Publishers 1980 · Zbl 0438.01006
[12] Scarf, H.: The core of ann person game. Econometrica35, 50-69 (1967) · Zbl 0183.24003
[13] Shapley, L.S.: On balanced games without side payments. In: Hu, T.C., Robinson, M. (eds.) Mathematical Program. Math. Res. Cent. Publ., vol. 30, (pp. 261-290). New York: Academic Press 1973 · Zbl 0267.90100
[14] Shih, M.-H., Lee, S.-N.: A combinatorial Lefschetz fixed-point formula. J. Comb. Theory, Ser. A61, 123-129 (1992) · Zbl 0771.55008
[15] Sperner, E.: Neuer Beweis f?r die Invarianz der Dimensionszahl und des Gebietes. Abh. Math. Semin. Univ. Hamburg6, 265-272 (1928) · JFM 54.0614.01
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