Excluding long paths

Abstract

Ding (1992) proved that for each integer ${m} \geqslant 0$, and every infinite sequence of finite simple graphs $G_1, G_2, \ldots$, if none of these graphs contains a path of length ${m}$ as a subgraph, then there are indices $i < j$ such that $G_i$ is isomorphic to an induced subgraph of $G_j$. We generalise this result to infinite graphs, possibly with parallel edges and loops.