Unavoidable Induced Subgraphs of Large 2-Connected Graphs

Ramsey proved that for every positive integer \(n\) , every sufficiently large graph contains an induced \(K_n\) or \(\overline{K_n}\) . 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 \(K_n\) , \(K_{1,n}\) , or \(P_n\) . 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: \(K_n\) , a subdivision of \(K_{2,n}\) , a subdivision of \(K_{2,n}\) with an edge between the two vertices of degree \(n\) , and a well-defined structure similar to a ladder.