Unavoidable Structures in Large and Infinite Graphs

In this work, we present results on the unavoidable structures in large connected and large 2-connected graphs. For the relation of induced subgraphs, Ramsey proved that for every positive integer r, every sufficiently large graph contains as an induced subgraph either Kr or Kr. It is well known that, for every positive integer r, every sufficiently large connected graph contains an induced subgraph isomorphic to one of Kr, K1,r, and Pr. We prove an analogous result for 2-connected graphs. Similarly, for infinite graphs, every infinite connected graph contains an induced subgraph isomorphic to one of the following: an infinite complete graph, an infinite star, and a ray. Using some techniques from the finite result, we give the unavoidable induced subgraphs of infinite 2-connected graphs. We then shift our attention to the relation of bipartite minors defined in 2016 by Chudnovsky, Kalai, Nevo, Novik, and Seymour. For the relation of bipartite minors, we present the unavoidable substructures of both large connected and large 2-connected bipartite graphs