Random bipartite posets and extremal problems

C. Biró, P. Hamburger, H. A. Kierstead, A. Pór, W. T. Trotter, R. Wang

Research output: Contribution to journalArticle

Abstract

Previously, Erdős, Kierstead and Trotter [5] investigated the dimension of random height 2 partially ordered sets. Their research was motivated primarily by two goals: (1) analyzing the relativetightness of the Füredi–Kahn upper bounds on dimension in terms ofmaximum degree; and (2) developing machinery for estimating theexpected dimension of a random labeled poset on n points. For thesereasons, most of their effort was focused on the case 0 < p≤ 1 / 2.While bounds were given for the range 1 / 2 ≤ p< 1 , the relative accuracy of theresults in the original paper deteriorated as p approaches 1. Motivated by two extremal problems involving conditions that force aposet to contain a large standard example, we were compelled torevisit this subject, but now with primary emphasis on the range1 / 2 ≤ p< 1. Our sharpened analysis shows that as papproaches 1, the expected value of dimension increases andthen decreases, answering in the negative a question posed in the original paper.Along the way, we apply inequalities of Talagrand and Janson,establish connections with latin rectangles and the Euler product function,and make progress on both extremal problems.

Original languageEnglish (US)
Pages (from-to)618-646
Number of pages29
JournalActa Mathematica Hungarica
Volume161
Issue number2
DOIs
StatePublished - Aug 1 2020

Keywords

  • bipartite poset
  • dimension
  • poset
  • standard example

ASJC Scopus subject areas

  • Mathematics(all)

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    Biró, C., Hamburger, P., Kierstead, H. A., Pór, A., Trotter, W. T., & Wang, R. (2020). Random bipartite posets and extremal problems. Acta Mathematica Hungarica, 161(2), 618-646. https://doi.org/10.1007/s10474-020-01049-y