Unavoidable induced subgraphs of large and infinite $2$-edge-connected graphs

In 1930, Ramsey proved that every large graph contains either a large clique or a large edgeless graph as an induced subgraph. It is well known that every large connected graph contains a long path, a large clique, or a large star as an induced subgraph. Recently Allred, Ding, and Oporowski presented the unavoidable large induced subgraphs for large and infinite $2$-connected graphs. The $2$-edge-connected (sometimes called bridgeless) graphs form an important class between connected graphs and $2$-connected graphs. In this paper we prove the existence of ubiquitous structures in $2$-edge-connected graphs known as chains of pinched super-clean ladders, and incorporate these into a presentation of the unavoidable large induced subgraphs for large and infinite $2$-edge-connected graphs. As consequences we obtain results on unavoidable large subgraphs, topological minors, minors, induced topological minors, induced minors, and Eulerian subgraphs in large and infinite $2$-edge-connected graphs. When appropriate we extend our results to multigraphs.