Trees of tangles in infinite separation systems

Abstract

We present infinite analogues of our splinter lemma for constructing nested sets of separations. From these we derive several tree-of-tangles-type theorems for infinite graphs and infinite abstract separation systems.

A tree-of-tangles theorem for infinite tangles

“…Very recently, Carmesin, Hamann and Miraftab showed a canonical version of this theorem; see their paper [21]. We should also mention the work by Dunwoody and Krön [22] and the work by Elbracht, Kneip and Teegen [9].…”

“…Elbracht, Kneip and Teegen use our Corollary 6.1 in their paper [9] to give a relatively short proof of Carmesin's tree-of-tangles theorem.…”

“…All of the four applications are, in fact, applications of Theorems 5.10 and 5.12. Elbracht, Kneip and Teegen [9] use the following corollary of our two theorems: Corollary 6.1 Suppose that Y is a principal 4 collection of vertex sets of G. Then there is a function K assigning to each X 2 Y a subset KðX Þ C X (the set C X consists of the components of G À X whose neighbourhoods are precisely equal to X) that misses at most one component from C X , such that the collection…”