Abstract
LetS(q, d) be the maximal numberv such that, for every general position linear maph: Δ(q−1)(d+1) →R d, there exist at leastv different collections {Δt1, ..., Δt q} of disjoint faces of Δ(q−1)(d+1) with the property thatf(Δt1) ∩ ... ∩f(Δt q) ≠ Ø. Sierksma's conjecture is thatS(q, d)=((q−1)!)d. The following lower bound (Theorem 1) is proved assuming thatq is a prime number:
Using the same technique we obtain (Theorem 2) a lower bound for the number of different splittings of a “generic” necklace.
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References
N. Alon, Splitting necklaces,Adv. in Math. 63 (1987), 247–253.
N. Alon, Some recent combinatorial applications of Borsuk-type theorem, inAlgebraic, Extremal and Metric Combinatorics, M. Deza, P. Frankl, and D. G. Rosenberg (eds.), Cambridge University Press, Cambridge, pp. 1–12.
I. Bárány, Geometric and combinatorial applications of Borsuk's theorem, A survey, preprint.
I. Bárány, S. B. Shlosman, and A. Szücs, On a topological generalization of a theorem of Tverberg,J. London Math. Soc. (2)23 (1981), 158–164.
A. Björner, Shellable and Cohen-Macaulay partially ordered sets,Trans. Amer. Math. Soc. 260 (1980), 159–183.
A. Björner, Topological methods, inHandbook of Combinatorics, R. Graham, M. Grötschel, and L. Lovász, (eds.), in press.
A. Björner and M. Wachs, On lexicographically shellable posets,Trans. Amer. Math. Soc. 277 (1983), 323–341.
L. Budach, B. Graw, C. Meinel, and S. Waack,Algebraic and Topological Properties of Finite Partially Ordered Sets, Teubner, Leipzig, 1988.
M. A. Krasnoselsky and P. P. Zabrejko,Geometritshskie zatatshi nelinejnogo analiza, Nauka, Moscow, 1975.
M. Ozaydin, to appear.
J. R. Reay, Open problems around Radon's theorem, inConvexity and Related Problems in Combinatorial Geometry, D. C. Kay and M. Breen (eds.), Marcel Dekker, New York, 1982, pp. 151–171.
K. S. Sarkaria, Sierksma's “Dutch cheese problem”, preprint.
K. S. Sarkaria, Kuratowski complexes,Topology, to appear.
T. tom Dieck,Transformation Groups, Studies in mathematics, Vol. 8, De Gruyter.
H. Tverberg, A generalization of Radon's theorem,J. London Math. Soc. 41, 123–128.
R. T. Živaljević and S. T. Vrećica, The colored Tverberg's problem and complexes of injective functions,J. Combin. Theory Ser. A, to appear.
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Work on this paper was supported by the Mathematical Institute, Belgrade, through the Serbian Science Foundation under Grant No. 0401D.
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Vučić, A., Živaljević, R.T. Note on a conjecture of sierksma. Discrete Comput Geom 9, 339–349 (1993). https://doi.org/10.1007/BF02189327
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DOI: https://doi.org/10.1007/BF02189327