UNAVOIDABLE INDUCED SUBGRAPHS OF LARGE 2-CONNECTED GRAPHS

Ramsey proved that for every positive integer n, every sufficiently large graph contains an induced Kn or Kn. Among the many extensions of Ramsey's theorem, there is an analogue for connected graphs: For every positive integer n, every sufficiently large connected graph contains an induced Kn, K1,n, or Pn. In this paper, we establish an analogue for 2-connected graphs. In particular, we prove that for every integer exceeding two, every sufficiently large 2-connected graph contains one of the following as an induced subgraph: Kn, a subdivision of K2,n, a subdivision of K2,n with an edge between the two vertices of degree n, and a well-defined structure similar to a ladder.