What structure/randomness dichotomies can be found across families of infinite structures? How precisely can we pinpoint the interactions of randomness and freedom vs structure and control inside a structure? Model theory has long investigated these questions. It turns out that model theoretic theorems about infinite objects can have strong consequences for finite objects. The talk will expand on these questions and some motivating consequences from the speaker's work (joint with S. Shelah), including characterization of the existence of irregular pairs in Szemeredi's regularity lemma and proofs of instances of Erdos-Hajnal.