Abstract
Sperner’s bound on the size of an antichain in the lattice P(S) of subsets of a finite set S has been generalized in three different directions: by Erdős to subsets of P(S) in which chains contain at most r elements; by Meshalkin to certain classes of compositions of S; by Griggs, Stahl, and Trotter through replacing the antichains by certain sets of pairs of disjoint elements of P(S). We unify these three bounds with a common generalization. We similarly unify their accompanying LYM inequalities. Our bounds do not in general appear to be the best possible.
Research supported by National Science Foundation grant DMS-0070729.
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References
I. Anderson, Combinatorics of Finite Sets, Clarendon Press, Oxford, 1987.
M. Beck and T. Zaslavsky, A shorter, simpler, stronger proof of the Meshalkin-Hochberg-Hirsch bounds on componentwise antichains, J. Combin. Theory Ser. A, 100 (2002), 196–199.
B. Bollobás, On generalized graphs. Acta Math. Acad. Sci. Hung., 16 (1965), 447–452.
P. Erdős, On a lemma of Littlewood and Offord, Bull. Amer. Math. Soc., 51 (1945), 898–902.
J. R. Griggs, J. Stahl and W. T. Trotter, A Sperner theorem on unrelated chains of subsets, J. Combinatorial Theory Ser. A, 36 (1984), 124–127.
M. Hochberg and W. M. Hirsch, Sperner families, s-systems, and a theorem of Meshalkin, Ann. New York Acad. Sci., 175 (1970), 224–237.
D. A. Klain and G.-C. Rota, Introduction to Geometric Probability, Cambridge University Press, Cambridge, Eng., 1997.
D. A. Lubell, A short proof of Sperner’s theorem, J. Combinatorial Theory, 1 (1966), 209–214.
L. D. Meshalkin, Generalization of Sperner’s theorem on the number of subsets of a finite set (in Russian), Teor. Verojatnost. i Primenen, 8 (1963), 219–220. English trans.: Theor. Probability Appl., 8 (1963), 203–204.
G.-C. Rota and L. H. Harper, Matching theory, an introduction, in: P. Ney, ed., Advances in Probability and Related Topics, Vol. 1, pp. 169–215, Marcel Dekker, New York, 1971.
E. Sperner, Ein Satz über Untermengen einer endlichen Menge, Math. Z., 27 (1928), 544–548.
K. Yamamoto, Logarithmic order of free distributive lattice, J. Math. Soc. Japan, 6 (1954), 343–353.
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Dedicated to the memories of Pál Erdős and Lev Meshalkin
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© 2006 János Bolyai Mathematical Society and Springer Verlag
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Beck, M., Wang, X., Zaslavsky, T. (2006). A Unifying Generalization of Sperner’s Theorem. In: Győri, E., Katona, G.O.H., Lovász, L., Fleiner, T. (eds) More Sets, Graphs and Numbers. Bolyai Society Mathematical Studies, vol 15. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-32439-3_1
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DOI: https://doi.org/10.1007/978-3-540-32439-3_1
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-32377-8
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