On a problem of R. Halin concerning infinite graphs

Abstract

Let α be an arbitrary cardinal ≠ 0 and let A be the graph that arises from a disjoint one-way infinite paths by identifying their initial vertices. For this graph A we show the following. (∗) Let G be an arbitrary graph which contains for every positive integer n a system of n disjoint graphs each isomorphic to A; then G also contains infinitely many disjoint subgraphs each isomorphic to A. This sharpens two theorems of Halin, who proved the corresponding result for the case that A is a one-way or two-way infinite path. Furthermore we show that (∗) holds, if A is an arbitrary countable tree with a finite diameter.