It is well known that, for every positive integer $r$, every sufficiently large connected graph contains an induced subgraph isomorphic to one of the following: a large complete graph, a large star, and a long path.  We prove an analogous result for 2-connected graphs.  In particular, we show that every sufficiently large 2-connected graph contains an induced subgraph isomorphic to one of the following: a complete graph, a subdivision of $K_{2,r}$ with possibly an edge joining the two vertices of degree $r$, and a graph that has a well-described ladder structure.

 

This is joint work with Guoli Ding and Bogdan Oporowski.