Weak Randomness in Graphons and Theons

ABSTRACT Call a hereditary family of graphs strongly persistent if there exists a graphon such that in all subgraphons of , is precisely the class of finite graphs that have positive density in . Our first result is a complete characterization of the hereditary families of graphs that are strongly persistent as precisely those that are closed under substitutions. We call graphons with the self‐similarity property above weakly random. A hereditary family is said to have the weakly random Erdős–Hajnal property () if every graphon that is a limit of graphs in has a weakly random subgraphon. Among families of graphs that are closed under substitutions, we completely characterize the families that belong to as those with “few” prime graphs. We also extend some of the results above to structures in finite relational languages by using the theory of theons.