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Journal of Combinatorial Optimization

, Volume 25, Issue 4, pp 587–596 | Cite as

The partition method for poset-free families

  • Jerrold R. Griggs
  • Wei-Tian Li
Article

Abstract

Given a finite poset P, let \({\rm La}(n,P)\) denote the largest size of a family of subsets of an n-set that does not contain P as a (weak) subposet. We employ a combinatorial method, using partitions of the collection of all full chains of subsets of the n-set, to give simpler new proofs of the known asymptotic behavior of \({\rm La}(n,P)\), as n, when P is the r-fork \(\mathcal {V}_{r}\), the four-element N poset \(\mathcal {N}\), and the four-element butterfly-poset \(\mathcal {B}\).

Keywords

Combinatorics of partially ordered sets Extremal set theory Sperner theory Lubell function 

References

  1. Bukh B (2009) Set families with a forbidden poset. Electron J Comb 16:R142 MathSciNetGoogle Scholar
  2. Carroll T, Katona GOH (2008) Bounds on maximal families of sets not containing three sets with \(A\cup B\subset C, A \not\subset B\). Order 25:229–236 MathSciNetCrossRefMATHGoogle Scholar
  3. De Bonis A, Katona GOH (2007) Largest families without an r-fork. Order 24:181–191 MathSciNetCrossRefMATHGoogle Scholar
  4. De Bonis A, Katona GOH, Swanepoel KJ (2005) Largest family without ABCD. J Comb Theory, Ser A 111:331–336 CrossRefMATHGoogle Scholar
  5. Erdős P (1945) On a lemma of Littlewood and Offord. Bull Am Math Soc 51:898–902 CrossRefGoogle Scholar
  6. Greene C, Kleitman DJ (1978) Proof techniques in the theory of finite sets. In: Rota GC (ed) Studies in combinatorics. MAA studies in mathematics, vol 17. MAA, Providence, pp 22–79 Google Scholar
  7. Griggs JR, Katona GOH (2008) No four subsets forming an N. J Comb Theory, Ser A 115:677–685 MathSciNetCrossRefMATHGoogle Scholar
  8. Griggs JR, Lu L (2009) On families of subsets with a forbidden subposet. Comb Probab Comput 18:731–748 MathSciNetCrossRefMATHGoogle Scholar
  9. Griggs JR, Li W-T, Lu L (2012) Diamond-free families. J Comb Theory, Ser. A 119:310–322 MathSciNetCrossRefMATHGoogle Scholar
  10. Katona GOH, Tarján TG (1983) Extremal problems with excluded subgraphs in the n-cube. In: Borowiecki M, Kennedy JW, Sysło MM (eds) Graph theory, Łagów, 1981. Lecture notes in math, vol 1018. Springer, Berlin, pp 84–93 CrossRefGoogle Scholar
  11. Lubell D (1966) A short proof of Sperner’s lemma. J Comb Theory 1:299 MathSciNetCrossRefMATHGoogle Scholar
  12. Sperner E (1928) Ein Satz über Untermegen einer endlichen Menge. Math Z 27:544–548 MathSciNetCrossRefMATHGoogle Scholar
  13. Stanley RP (1997) Enumerative combinatorics, vol 1. Cambridge University Press, Cambridge CrossRefMATHGoogle Scholar
  14. Thanh HT (1998) An extremal problem with excluded subposets in the boolean lattice. Order 15:51–57 MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of South CarolinaColumbiaUSA
  2. 2.Institute of MathematicsAcademia SinicaTaipeiTaiwan

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