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, 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
Article

Abstract

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].

Keywords

Posets Rank preserving copy P-free Extremal number 

Notes

Acknowledgments

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.

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Copyright information

© The Author(s) 2019

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), 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

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