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Set Families With a Forbidden Induced Subposet


For each poset H whose Hasse diagram is a tree of height k, we show that the largest size of a family of subsets of [n]={1,. . ., n} not containing H as an induced subposet is asymptotic to . This extends a result of Bukh [1], which in turn generalizes several known results including Sperner's theorem.

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[1]Bukh, B. (2009) Set families with a forbidden subposet. Electron. J. Combin. 16 #R142.
[2]Carroll, T. and Katona, G. O. H. (2008) Bounds on maximal families of sets not containing three sets with ABC, A∉⊂B. Order 25 229236.
[3]De Bonis, A. and Katona, G. O. H. (2007) Largest families without an r-fork. Order 24 181191.
[4]De Bonis, A., Katona, G. O. H. and Swanepoel, K. J. (2005) Largest families without ABCD. J. Combin. Theory Ser. A 111 331336.
[5]Erdős, P. (1945) On a lemma of Littlewood and Offord. Bull. Amer. Math. Soc. 51 898902.
[6]Griggs, J. and Lu, L. (2009) On families of subsets with a forbidden subposet. Combin. Probab. Comput. 18 731748.
[7]Sperner, E. (1928) Ein Satz über Untermengen einer endlichen Menge. Math. Z. 27 544548.
[8]Thanh, H. T. (1998) An extremal problem with excluded subposets in the Boolean lattice. Order 15 5157.
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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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