Introduction

Given a fixed graph H on t vertices, a typical graph G on n vertices contains many induced subgraphs isomorphic to H as n becomes large. Indeed, for the usual model of a random graph G* on n vertices (see [4]), in which potential edges are independently included or not each with probability ½, almost all such G* contain induced copies of H as n → ∞. Thus, if a large graph G contains no induced copy of H, it deviates from being ‘typical’ in a rather strong way. In this case, we would expect it to behave quite differently from random graphs in many other ways as well. That this in fact must happen follows from recent work of several authors, e.g., see Chung, Graham & Wilson [5] and Thomason [7], [8]. In this paper we initiate a quantitative study of how various deviations of randomness are related. The particular property we investigate (‘uniform edge density for half sets’ – see Section 3) is just one of many which might have been selected and for which the same kind of analysis could be carried out.

This work also shares a common philosophy with several recent papers of Alon & Bollobás [1] and Erdős & Hajnal [6], which investigate the structure of graphs which have an unusually small number of non-isomorphic induced subgraphs. This is a strong restriction and such graphs must have very large subgraphs which are (nearly) complete or independent.