Random Event Structures

In a line of recent development, probabilistic constructions of universal, homogeneous objects have been provided in various categories of ordered structures, such as causal sets, bifinite domains, and countable partial orders. These constructions have been shown to produce objects with the desired properties with probability 1 in an appropriately defined measure space. A common strategy for these constructions is successive point-wise extension of an existing finite structure, with decisions on the relationships between the newly added point and the existing structure made according to well-specified probabilistic choices. This strategy is a departure from (and understandably so due to the increased complexity) the original one for random graphs where a universal homogeneous countable graph is constructed with probability 1 in a single step (i.e., a single round of countably many probabilistic choices made independently). It would be interesting to see which of the categories studied more recently may admit such “one-step” constructions. The main focus of this paper is a new strategy, consisting of a single round of countably many probabilistic choices made independently, for the construction of a universal, homogeneous prime event structure. The intuition that the one-round construction is desirable has a similar flavor to a more general setting in e.g. Calculus/Real Analysis. When taking limits, iterative stepby-step processes are usually given, but a set of machineries was invented to determine the limit, i.e., achieving a “one-round” direct and explicit description of the limit.