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].
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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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Gerbner, D., Methuku, A., Nagy, D.T. et al. Forbidding Rank-Preserving Copies of a Poset. Order 36, 611–620 (2019). https://doi.org/10.1007/s11083-019-09484-5
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DOI: https://doi.org/10.1007/s11083-019-09484-5