On the typical structure of graphs not containing a fixed vertex-critical subgraph

This work studies the typical structure of sparse (Formula presented.) -free graphs, that is, graphs that do not contain a subgraph isomorphic to a given graph (Formula presented.). Extending the seminal result of Osthus, Prömel, and Taraz that addressed the case where (Formula presented.) is an odd cycle, Balogh, Morris, Samotij, and Warnke proved that, for every (Formula presented.), the structure of a random (Formula presented.) -free graph with (Formula presented.) vertices and (Formula presented.) edges undergoes a phase transition when (Formula presented.) crosses an explicit (sharp) threshold function (Formula presented.). They conjectured that a similar threshold phenomenon occurs when (Formula presented.) is replaced by any strictly 2-balanced, edge-critical graph (Formula presented.). In this paper, we resolve this conjecture. In fact, we prove that the structure of a typical (Formula presented.) -free graph undergoes an analogous phase transition for every (Formula presented.) in a family of vertex-critical graphs that includes all edge-critical graphs.