, Volume 36, Issue 3, pp 611–620 | Cite as

Forbidding Rank-Preserving Copies of a Poset

  • Dániel GerbnerEmail author
  • Abhishek Methuku
  • Dániel T. Nagy
  • Balázs Patkós
  • Máté Vizer
Open Access


The maximum size, La(n,P), of a family of subsets of [n] = {1,2,...,n} without containing a copy of P as a subposet, has been extensively studied. Let P be a graded poset. We say that a family \({\mathcal F}\) of subsets of [n] = {1,2,...,n} contains a rank-preserving copy of P if it contains a copy of P such that elements of P having the same rank are mapped to sets of same size in \({\mathcal F}\). The largest size of a family of subsets of [n] = {1,2,...,n} without containing a rank-preserving copy of P as a subposet is denoted by Larp(n,P). Clearly, La(n,P) ≤ Larp(n,P) holds. In this paper we prove asymptotically optimal upper bounds on Larp(n,P) for tree posets of height 2 and monotone tree posets of height 3, strengthening a result of Bukh in these cases. We also obtain the exact value of \(La_{rp}(n,\{Y_{h,s},Y_{h,s}^{\prime }\})\) and \(La(n,\{Y_{h,s},Y_{h,s}^{\prime }\})\), where Yh,s denotes the poset on h + s elements \(x_{1},\dots ,x_{h},y_{1},\dots ,y_{s}\) with \(x_{1}<\dots <x_{h}<y_{1},\dots ,y_{s}\) and \(Y^{\prime }_{h,s}\) denotes the dual poset of Yh,s, thereby proving a conjecture of Martin et. al. [10].


Posets Rank preserving copy P-free Extremal number 



Open access funding provided by MTA Alfréd Rényi Institute of Mathematics (MTA RAMKI). DG’s research supported by the János Bolyai Research Fellowship of the Hungarian Academy of Sciences and the National Research, Development and Innovation Office – NKFIH under the grant K 116769.

DTN’s research supported by the ÚNKP-17-3 New National Excellence Program of the Ministry of Human Capacities and by National Research, Development and Innovation Office – NKFIH under the grant K 116769.

BP’s research supported by the National Research, Development and Innovation Office – NKFIH under the grants SNN 116095 and K 116769.

MV’s research supported by the National Research, Development and Innovation Office – NKFIH under the grant SNN 116095.


  1. 1.
    Boehnlein, E., Jiang, T.: Set families with a forbidden induced subposet. Comb. Probab. Comput. 21(4), 496–511 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bukh, B.: Set families with a forbidden subposet. Electron. J. Comb. 16(1), R142 (2009)MathSciNetzbMATHGoogle Scholar
  3. 3.
    De Bonis, A., Katona, G.O., Swanepoel, K.J.: Largest family without ABCD. Journal of Combinatorial Theory, Series A 111(2), 331–336 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Erdös, P.: On a lemma of Littlewood and Offord. Bull. Am. Math. Soc. 51(12), 898–902 (1945)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Griggs, J.R., Li, W.-T.: The partition method for poset-free families. J. Comb. Optim. 25(4), 587–596 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Griggs, J.R., Li, W.-T.: Progress on poset-free families of subsets. In: Recent Trends in Combinatorics, pp. 317–338. Springer (2016)Google Scholar
  7. 7.
    Griggs, J.R., Lu, L.: On families of subsets with a forbidden subposet. Comb. Probab. Comput. 18(05), 731–748 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Katona, G., Tarján, T.G.: Extremal problems with excluded subgraphs in the n-cube. In: Graph Theory, pp. 84–93. Springer (1983)Google Scholar
  9. 9.
    Kleitman, D.: A conjecture of Erdős-Katona on commensurable pairs among subsets of an n-set. In: Theory of Graphs, Proc. Colloq. held at Tihany, pp. 187–207. Hungary, (1966)Google Scholar
  10. 10.
    Martin, R.R., Methuku, A., Uzzell, A., Walker, S.: A discharging method for forbidden subposet problems. arXiv:1710.05057
  11. 11.
    Methuku, A., Pálvölgyi, D.: Forbidden hypermatrices imply general bounds on induced forbidden subposet problems Combinatorics, Probability and Computing, 1–10 (2017)Google Scholar
  12. 12.
    Methuku, A., Tompkins, C.: Exact forbidden subposet results using chain decompositions of the cycle. Electron. J. Comb. 22(4), P4–29 (2015)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Nagy, D.T.: Forbidden subposet problems with size restrictions. J. Comb. Theory, Ser. A 155, 42–66 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Patkós, B.: Induced and non-induced forbidden subposet problems. Electron. J. Comb. 22(1), P1–30 (2015)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Sperner, E.: Ein Satz über Untermengen einer endlichen Menge. Math Z 27(1), 585–592 (1928)zbMATHCrossRefGoogle Scholar
  16. 16.
    Thanh, H.T.: An extremal problem with excluded subposet in the boolean lattice. Order 15(1), 51–57 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Tompkins, C., Wang, Y.: On an extremal problem involving a pair of forbidden posets. arXiv:1710.10760

Copyright information

© The Author(s) 2019

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Dániel Gerbner
    • 1
    Email author
  • Abhishek Methuku
    • 2
  • Dániel T. Nagy
    • 1
  • Balázs Patkós
    • 1
  • Máté Vizer
    • 1
  1. 1.Alfréd Rényi Institute of MathematicsHungarian Academy of SciencesBudapestHungary
  2. 2.Department of MathematicsCentral European UniversityBudapestHungary

Personalised recommendations