Summary
There are two central themes to research involving applications of probabilistic methods to partially ordered sets. The first of these can be described as the study of random partially ordered sets. Among the specific models which have been studied are: random labelled posets; random t-dimensional posets; and the transitive closure of random graphs. A second theme concentrates on the adaptation of random methods so as to be applicable to general partially ordered sets. In this paper, we concentrate on the second theme. Among the topics we discuss are fibers and co-fibers; the dimension of subposets of the subset lattice; the dimension of posets of bounded degree; and fractional dimension. This last topic leads to a discussion of Ramsey theoretic questions for probability spaces.
1991 Mathematics Subject Classification. 06A07, 05C35.
Key words and phrases. Partially ordered set, poset, graph, random methods, dimension, fractional dimension, chromatic number Research supported in part by the Office of Naval Research.
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This paper is dedicated to Paul Erdős with appreciation for his impact on mathematics and the lives of mathematicians all over the world.
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Trotter, W.T. (2013). Applications of the Probabilistic Method to Partially Ordered Sets. In: Graham, R., Nešetřil, J., Butler, S. (eds) The Mathematics of Paul Erdős II. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7254-4_20
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