Abstract
Let f(n) be the largest integer such that every poset on n elements has a 2-dimensional subposet on f(n) elements. What is the asymptotics of f(n)? It is easy to see that f(n) = n 1/2. We improve the best known upper bound and show f(n) = O (n 2/3). For higher dimensions, we show \(f_{d}(n)=\O \left (n^{\frac {d}{d + 1}}\right )\), where f d (n) is the largest integer such that every poset on n elements has a d-dimensional subposet on f d (n) elements.
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Acknowledgements
We send thanks to Wojciech Samotij and Dömötör Pálvölgyi for pointing us to useful references.
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Grzegorz Guśpiel was partially supported by the Polish Ministry of Science and Higher Education grant DI2013 000443. Piotr Micek was partially supported by the National Science Center of Poland under grant no. 2015/18/E/ST6/00299. Adam Polak was partially supported by the Polish Ministry of Science and Higher Education program “Diamentowy Grant”.
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Guśpiel, G., Micek, P. & Polak, A. On an Extremal Problem for Poset Dimension. Order 35, 489–493 (2018). https://doi.org/10.1007/s11083-017-9444-1
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DOI: https://doi.org/10.1007/s11083-017-9444-1