Combinatorial and number-theoretic properties of generic reals

Abstract

We discuss some properties of Cohen and random reals. We show that they belong to any definable partition regular family, and hence they satisfy most "largeness" properties studied in Ramsey theory. We determine their position in the Mahler's classification of the reals and using it, we get some information about Liouville numbers. We also show that they are wild in the sense of o-minimality, i.e., they define the set of integers.