# Box-spaces and random partial orders

## Abstract

Winkler [2] studied random partially ordered sets, defined by taking n points at random in [0, 1]d, with the order on these points given by the restriction of the order on [0, 1]d. Bolloblis 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) [0, 1]d 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 d 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 n random points in a box-space.) We extend the results of Bollobas and Winkler to these random posets. In particular we show that, for a box-space X of dimension d, there is a constant mx such that the length of a longest chain tends to m × n l/d in probability. © 1991 American Mathematical Society.

## Publication Title

Transactions of the American Mathematical Society

## Recommended Citation

Bollobás, B., & Brightwell, G.
(1991). Box-spaces and random partial orders.* Transactions of the American Mathematical Society**, 324* (1), 59-72.
https://doi.org/10.1090/S0002-9947-1991-0986685-9