Typical Structure of Hereditary Graph Families. II. Exotic Examples

A graph (Formula presented.) is (Formula presented.) -free if it does not contain an induced subgraph isomorphic to (Formula presented.). The study of the typical structure of (Formula presented.) -free graphs was initiated by Erdős, Kleitman, and Rothschild (1976), who have shown that almost all (Formula presented.) -free graphs are bipartite. Since then the typical structure of (Formula presented.) -free graphs has been determined for several families of graphs (Formula presented.), including complete graphs, trees, and cycles. Recently, Reed and Scott proposed a conjectural description of the typical structure of (Formula presented.) -free graphs for all graphs (Formula presented.), 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 (Formula presented.) -free graphs for graphs (Formula presented.) in this family. Using similar techniques, we construct an infinite family of graphs (Formula presented.) for which the maximum size of a homogenous set in a typical (Formula presented.) -free graph is sublinear in the number of vertices, answering a question of Loebl et al. (2010) and Kang et al. (2014).