Previously, Erdős, Kierstead and Trotter  investigated the dimension of random height 2 partially ordered sets. Their research was motivated primarily by two goals: (1) analyzing the relative tightness of the Füredi–Kahn upper bounds on dimension in terms of maximum degree; and (2) developing machinery for estimating the expected dimension of a random labeled poset on n points. For these reasons, most of their effort was focused on the case \(0<p\le 1/2\). While bounds were given for the range \(1/2\le p <1\), the relative accuracy of the results in the original paper deteriorated as p approaches 1.
Motivated by two extremal problems involving conditions that force a poset to contain a large standard example, we were compelled to revisit this subject, but now with primary emphasis on the range \(1/2\le p<1\). Our sharpened analysis shows that as p approaches 1, the expected value of dimension increases and then 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.
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The authors would like to thank Noga Alon, Alan Frieze, and Tomasz Łuczak for very helpful communications concerning second moment methods, the concept of defect, and Talagrand’s inequality. As noted previously, the heart of the proof of Lemma 3.5 was provided by Łuczak in a personal communication.
Dedicated to Endre Szemerdi on the occasion of his 80th birthday
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Biró, C., Hamburger, P., Kierstead, H.A. et al. Random bipartite posets and extremal problems. Acta Math. Hungar. (2020). https://doi.org/10.1007/s10474-020-01049-y
Key words and phrases
- bipartite poset
- standard example
Mathematics Subject Classification