Typical Structure of Hereditary Graph Families. II. Exotic Examples

ABSTRACT A graph is ‐free if it does not contain an induced subgraph isomorphic to . The study of the typical structure of ‐free graphs was initiated by Erdős, Kleitman, and Rothschild (1976), who have shown that almost all ‐free graphs are bipartite. Since then the typical structure of ‐free graphs has been determined for several families of graphs , including complete graphs, trees, and cycles. Recently, Reed and Scott proposed a conjectural description of the typical structure of ‐free graphs for all graphs , which extends all previously known results in the area. We construct an infinite family of graphs for which the Reed–Scott conjecture fails, and use the methods we developed in the prequel paper Norin and Yuditsky (2024) to describe the typical structure of ‐free graphs for graphs in this family. Using similar techniques, we construct an infinite family of graphs for which the maximum size of a homogenous set in a typical ‐free graph is sublinear in the number of vertices, answering a question of Loebl et al. (2010) and Kang et al. (2014).