Which subsets of an infinite random graph look random?

Infinite Subsets of Random Sets of Integers

There is an infinite subset of a Martin-Löf random set of integers that does not compute any Martin-Löf random set of integers. To prove this, we show that each real of positive effective Hausdorff dimension computes an infinite subset of a Martin-Löf random set of integers, and apply a result of Miller.

A note on orientations of the infinite random graph

We answer a question of P. Cameron’s by giving examples of 2א0 many non-isomorphic acyclic orientations of the infinite random graph with a topological ordering that do not have the pigeonhole property. Our examples also embed each countable linear ordering. A graph is n-existentially closed or n-e.c. if for each n-subset S of vertices, and each subset T of S (possibly empty), there is a vertex...

On Random Shuffles & Subsets

Four simple algorithms are presented for producing a random rearrangement of the values in an array X[1], X[2], ..., X[n] using a pseudorandom number generator. These range from one that is naive, inefficient, and biased to one that is elegeant, efficient, and unbiased. By taking the first k entries in X one obtains a random (ordered) subset of size k.

Quasi - Random Subsets of Z

Random generations of the countable random graph

We consider random processes more general than those considered by Erdös and Rényi for generating the countable random graph. It is proved that, in the category sense, almost all random processes we consider generate the countable random graph with probability 1. Under a weak boundedness assumption we give a criterion for the random processes which generate the countable random graph almost sur...