The Mathematics of Paul Erdős II pp 313-329 | Cite as

# Applications of the Probabilistic Method to Partially Ordered Sets

## 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.

## Keywords

Fractional Dimension Linear Order Chromatic Number Minimal Element Linear Extension## Bibliography

- 1.G. R. Brightwell, Models of random partial orders, in
*Surveys in Combinatorics 1993*, K. Walker, ed., 53–83.Google Scholar - 2.G. R. Brightwell, Graphs and partial orders,
*Graphs and Mathematics*, L. Beineke and R. J. Wilson, eds., to appear.Google Scholar - 3.G. R. Brightwell and E. R. Scheinerman, Fractional dimension of partial orders,
*Order***9**(1992), 139–158.Google Scholar - 4.R. P. Dilworth, A decomposition theorem for partially ordered sets,
*Ann. Math*.**51**(1950), 161–165.MathSciNetMATHCrossRefGoogle Scholar - 5.D. Duffus, H. Kierstead and W. T. Trotter, Fibres and ordered set coloring,
*J. Comb. Theory Series A***58**(1991) 158–164.Google Scholar - 6.D. Duffus, B. Sands, N. Sauer and R.Woodrow, Two coloring all two-element maximal antichains,
*J. Comb. Theory Series A***57**(1991) 109–116.Google Scholar - 7.B. Dushnik, Concerning a certain set of arrangements,
*Proc. Amer. Math. Soc*.**1**(1950), 788–796.MathSciNetMATHCrossRefGoogle Scholar - 8.P. Erdős, H. Kierstead and W. T. Trotter, The dimension of random ordered sets,
*Random Structures and Algorithms***2**(1991), 253–275.Google Scholar - 9.P. Erdős and L. Lovász, Problems and results on 3-chromatic hypergraphs and some related questions, in
*Infinite and Finite Sets*, A. Hajnal et al., eds., North Holland, Amsterdam (1975) 609–628.Google Scholar - 10.S. Felsner and W. T. Trotter, On the fractional dimension of partially ordered sets,
*Discrete Math*.**136**(1994), 101–117.MathSciNetMATHCrossRefGoogle Scholar - 11.P. C. Fishburn, Intransitive indifference with unequal indifference intervals,
*J. Math. Psych*.**7**(1970), 144–149.MathSciNetMATHCrossRefGoogle Scholar - 12.Z. Füredi and J. Kahn, On the dimensions of ordered sets of bounded degree,
*Order***3**(1986) 17–20.Google Scholar - 13.Z. Füredi, P. Hajnal,V. Rödl, and W. T. Trotter, Interval orders and shift graphs, in
*Sets, Graphs and Numbers*, A. Hajnal and V. T. Sos, eds., Colloq. Math. Soc. Janos Bolyai**60**(1991) 297–313.Google Scholar - 14.D. Kelly and W. T. Trotter, Dimension theory for ordered sets, in
*Proceedings of the Symposium on Ordered Sets*, I. Rival et al., eds., Reidel Publishing (1982), 171–212.Google Scholar - 15.H. A. Kierstead, The order dimension of 1-sets versus
*k*-sets, J. Comb. Theory Series A, to appear.Google Scholar - 16.D. J. Kleitman and G. Markovsky, On Dedekind’s problem: The number of isotone boolean functions, II,
*Trans. Amer. Math. Soc*.**213**(1975), 373–390.MathSciNetMATHGoogle Scholar - 17.
- 18.Z. Lonc and I. Rival, Chains, antichains and fibers,
*J. Comb. Theory Series A***44**(1987) 207–228.Google Scholar - 19.R. Maltby, A smallest fibre-size to poset-size ratio approaching 8/15,
*J. Comb. Theory Series A***61**(1992) 331–332.Google Scholar - 20.J. Spencer, Minimal scrambling sets of simple orders,
*Acta Math. Acad. Sci. Hungar*.**22**, 349–353.Google Scholar - 21.W. T. Trotter, Graphs and Partially Ordered Sets, in
*Selected Topics in Graph Theory II*, R. Wilson and L. Beineke, eds., Academic Press (1983), 237–268.Google Scholar - 22.W. T. Trotter, Problems and conjectures in the combinatorial theory of ordered sets,
*Annals of Discrete Math*.**41**(1989), 401–416.MathSciNetCrossRefGoogle Scholar - 23.W. T. Trotter,
*Combinatorics and partially Ordered Sets: Dimension Theory*, The Johns Hopkins University Press, Baltimore, Maryland (1992).MATHGoogle Scholar - 24.W. T. Trotter, Progress and new directions in dimension theory for finite partially ordered sets, in
*Extremal Problems for Finite Sets*, P. Frankl, Z. Füredi, G. Katona and D. Miklós, eds., Bolyai Soc. Math. Studies**3**(1994), 457–477.Google Scholar - 25.W. T. Trotter, Partially ordered sets, in
*Handbook of Combinatorics*, R. L. Graham, M. Grötschel, L. Lovász, eds., to appear.Google Scholar - 26.W. T. Trotter, Graphs and Partially Ordered Sets, to appear.Google Scholar
- 27.W. T. Trotter and P. Winkler, Ramsey theory and sequences of random variables, in preparation.Google Scholar