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Abstract: We form the random poset P(n; p) by including each subset of [n] = f1; : : : ; ng with probability p and ordering the subsets by inclusion. We investigate the length of the longest chain contained in P(n; p). For p e=n we obtain the limit distribution of this random variable. For smaller p we give thresholds for the existence of chains which imply that almost surely the length of P(n; p) is asymptotically n(log n log log 1=p)= log 1=p. (Update)
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...the next section. For results concerning the length of P(n; p) i.e. the cardinality of a longest chain, the reader is referred to [1]. The threshold function for small sublattices in P(n; p) are investigated in [2] which may be thought of as a modern sequel to R enyi s...
...a chain is a subset of P(n) whose elements are pairwise comparable. To make this argument work, we make heavy use of a technical result from [4]: condition (1) essentially guarantees the existence of many chains in P(n; p) whose cardinalities are close to n=r. The lower bound...
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2: Small sublattices in random subsets of Boolean lattices (context) - Kreuter - 1997
BibTeX entry: (Update)
Y. Kohayakawa, B. Kreuter, and D. Osthus. The length of random subsets of Boolean lattices. In preparation, 1998. http://citeseer.ist.psu.edu/kohayakawa98length.html More
@article{ kohayakawa00length,
author = "Yoshiharu Kohayakawa and Bernd Kreuter and Deryk Osthus",
title = "The length of random subsets of Boolean lattices",
journal = "Random Structures and Algorithms",
volume = "16",
number = "2",
pages = "177-194",
year = "2000",
url = "citeseer.ist.psu.edu/kohayakawa98length.html" }
Citations (may not include all citations):957 The probabilistic method (context) - Alon, Spencer - 1992
79 The theory of branching processes (context) - Harris - 1963
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4 Models of random partial orders (context) - Brightwell - 1993
4 the diameter and radius of random subgraphs of the cube (context) - Bollobs, Kohayakawa et al. - 1994
4 Small sublattices in random subsets of Boolean lattices (context) - Kreuter - 1998
3 rst-passage percolation and covering times for Richardson's .. (context) - Fill, Pemantle - 1993
3 On random subsets of a nite set (context) - Rnyi - 1961
3 The width of random subsets of Boolean lattices - Kohayakawa, Kreuter - 1998
3 Connectivity properties of random subgraphs of the cube (context) - Bollobs, Kohayakawa et al. - 1995
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1 Maximum antichains in random subsets of a nite set (context) - Osthus
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