Box-spaces and random partial orders

Authors:
Béla Bollobás and Graham Brightwell

Journal:
Trans. Amer. Math. Soc. **324** (1991), 59-72

MSC:
Primary 60D05; Secondary 06A07

DOI:
https://doi.org/10.1090/S0002-9947-1991-0986685-9

MathSciNet review:
986685

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Winkler [2] studied random partially ordered sets, defined by taking points at random in , with the order on these points given by the restriction of the order on . Bollobás and Winkler [1] gave several results on the height of such a random partial order. In this paper, we extend these results to a more general setting. We define a box-space to be, roughly speaking, a partially ordered measure space such that every two intervals of nonzero measure are isomorphic up to a scale factor. We give some examples of box-spaces, including (i) with the usual measure and order, and (ii) Lorentzian space-time with the order given by causality. We show that, for every box-space, there is a constant which behaves like the dimension of the space. In the second half of the paper, we study random partial orders defined by taking a Poisson distribution on a box-space. (This is of course essentially the same as taking random points in a box-space.) We extend the results of Bollobás and Winkler to these random posets. In particular we show that, for a box-space of dimension , there is a constant such that the length of a longest chain tends to in probability.

**[1]**B. Bollobás and P. Winkler,*The longest chain among random points in Euclidean space*, Proc. Amer. Math. Soc.**103**(1988), 347-353. MR**943043 (89k:60011)****[2]**P. Winkler,*Random orders*, Order**1**(1985), 317-331. MR**787544 (86j:06004)****[3]**-,*Connectedness and diameter for random orders of fixed dimension*, Order**2**(1985), 165-171. MR**815862 (87h:06004)****[4]**J. Justicz, E. Scheinerman, and P. Winkler,*Random intervals*, Amer. Math. Monthly (to appear). MR**1079974 (91m:60023)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
60D05,
06A07

Retrieve articles in all journals with MSC: 60D05, 06A07

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1991-0986685-9

Article copyright:
© Copyright 1991
American Mathematical Society