Collections of subsets with the Sperner property
Author:
Jerrold R. Griggs
Journal:
Trans. Amer. Math. Soc. 269 (1982), 575591
MSC:
Primary 05A05; Secondary 05C35, 06A10, 52A37
MathSciNet review:
637711
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Abstract: Let and , . Let be the subsets of which intersect , ordered by inclusion. Lih showed that has the Sperner property. Here it is shown that has several stronger properties. A nested chain decomposition is constructed for by bracketing. is shown to have the LYM property. A more general class of collections of subsets is studied: Let be partitioned into parts , let be subsets of , and let . Sufficient conditions on the are given for to be LYM, or at least Sperner, and examples are provided in which is not Sperner. Other results related to Sperner's theorem, the KruskalKatona theorem, and the LYM inequality are presented.
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 G. F. Clements, More on the generalized Macaulay theorem. II, Discrete Math. 18 (1977), 253264. MR 0457232 (56:15442)
 [2]
 E. R. Gansner, On the lattice of order ideals of an updown poset (preprint). MR 675856 (84b:06012)
 [3]
 C. Greene and D. J. Kleitman, On the structure of Sperner families, J. Combin. Theory Ser. A 20 (1976), 4168. MR 0398844 (53:2695)
 [4]
 , Proof techniques in the theory of finite sets, Studies in Combinatorics (G.C. Rota, editor), Math. Assoc. Amer., 1978, pp. 2279. MR 513002 (80a:05006)
 [5]
 , Strong versions of Sperner's theorem, J. Combin. Theory Ser. A 20 (1976), 8088. MR 0389608 (52:10439)
 [6]
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 [7]
 , On chains and Sperner families in ranked posets, J. Combin. Theory Ser. A 28 (1980), 156168. MR 563553 (81e:05006)
 [8]
 J. R. Griggs, M. Saks and D. Sturtevant, On chains and Sperner families in ranked posets. II (to appear).
 [9]
 L. H. Harper, The morphology of partially ordered sets, J. Combin. Theory 17 (1974), 4458. MR 0366762 (51:3008)
 [10]
 W. N. Hsieh and D. J. Kleitman, Normalized matching in direct products of partial orders, Stud. Appl. Math. 52 (1973), 285289. MR 0404064 (53:7870)
 [11]
 G. O. H. Katona, A theorem of finite sets, Proc. Tihany Conf., 1966, Budapest, 1968.
 [12]
 D. J. Kleitman, Extremal hypergraph properties, Surveys in Combinatorics, Proc. 7th British Comb. Conf. (B. Bollobás, editor), London Math. Soc. Lecture Note Series no. 38, 1979.
 [13]
 , On an extremal property of antichains in partial orders: the LYM property and some of its implications and applications, Combinatorics (M. Hall and J. H. van Lint, editors), Math. Centre Tracts, Amsterdam, vol. 55, 1974, pp. 7790. MR 0360379 (50:12829)
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 J. Kruskal, The number of simplices in a complex, Math. Optimization Techniques, Univ. of California Press, Berkeley and Los Angeles, 1963, pp. 251278. MR 0154827 (27:4771)
 [15]
 K.W. Lih, Sperner families over a subset (preprint). MR 583957 (81j:05004)
 [16]
 D. Lubell, A short proof of Sperner's theorem, J. Combin. Theory 1 (1966), 299. MR 0194348 (33:2558)
 [17]
 R. A. Proctor, M. Saks and D. Sturtevant, Product partial orders with the Sperner property (to appear). MR 566433 (81f:06005)
 [18]
 E. Sperner, Ein Satz über Untermengen einer endlichen Menge, Math. Z. 27 (1928), 544548. MR 1544925
 [19]
 A. Wareham, Ph.D. dissertation, University of California, Riverside, 1980.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198206377112
PII:
S 00029947(1982)06377112
Article copyright:
© Copyright 1982 American Mathematical Society
