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Large subposets with small dimension. (English) Zbl 1336.06003
Summary: Dorais asked for the maximum guaranteed size of a subposet with dimension at most \(d\) of an \(n\)-element poset. A lower bound of order \(\sqrt n\) was found by Goodwillie. We provide a sublinear upper bound for each \(d\). For \(d=2\), our bound is \(n^{0.8295}\).

MSC:
06A07 Combinatorics of partially ordered sets
Software:
MathOverflow
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References:
[1] Dorais, F.G.: Subposets of small dimension http://dorais.org/archives/656 (updated Feb. 18, 2012, retrieved Feb. 17, 2014) (2014)
[2] Dorais, F.G.: Subposets of small Dushnik-Miller dimension http://mathoverflow.net/questions/29169 (updated Sept. 12, 2010, retrieved Feb. 17, 2014) (2014)
[3] Erdős, P.: Some new applications of probability methods to combinatorial analysis and graph theory. In: Proceedings of the Fifth Southeastern Conference on Combinatorics, Graph Theory and Computing, pp 39-51 (1974)
[4] Goodwillie, T.: http://mathoverflow.net/questions/29570 (updated June 28, 2013, retrieved Feb. 17, 2014) (2014)
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