Tree sets are abstract structures that can be used to model various tree-shaped objects in combinatorics. Finite tree sets can be represented by finite graph-theoretical trees. We extend this representation theory to infinite tree sets. First we characterise those tree sets that can be represented by tree sets arising from infinite trees; these are precisely those tree sets without a chain of order type ω + 1. Then we introduce and study a topological generalisation of infinite trees which can have limit edges, and show that every infinite tree set can be represented by the tree set admitted by a suitable such tree-like space.
Bowler, N., Carmesin, J., Christian, R.: Infinite graphic matroids. Combinatorica 38(2), 305–339 (2018). https://doi.org/10.1007/s00493-016-3178-3. MR3800843
Diestel, R.: Abstract separation systems. Order 35(1), 157–170 (2018). https://doi.org/10.1007/s11083-017-9424-5. MR3774512
Diestel, R.: Tree sets. Order 35(1), 171–192 (2018). https://doi.org/10.1007/s11083-017-9425-4. MR3774513
Diestel, R., Oum, S.: Tangle-tree duality in abstract separation systems, arXiv:1701.02509 (2017)
Diestel, R., Oum, S.: Tangle-tree duality: in graphs, matroids and beyond. Combinatorica 39(4), 879–910 (2019). https://doi.org/10.1007/s00493-019-3798-5. MR4015355
Diestel, R., Whittle, G.: Tangles and the Mona Lisa, arXiv:1603.06652 (2016)
Robertson, N., Seymour, P.D.: Graph minors. X. Obstructions to tree-decomposition. J. Combin. Theory Ser. B 52(2), 153–190 (1991). https://doi.org/10.1016/0095-8956(91)90061-N. MR1110468
Robertson, N., Seymour, P., Thomas, R.: Excluding infinite minors. Discrete Math. 95(1-3), 303–319 (1991). https://doi.org/10.1016/0012-365X(91)90343-Z. Directions in infinite graph theory and combinatorics (Cambridge, 1989), MR1141945
Seymour, P.D., Thomas, R.: Graph searching and a min-max theorem for tree-width. J. Combin. Theory Ser. B 58(1), 22–33 (1993). https://doi.org/10.1006/jctb.1993.1027. MR1214888
Thomassen, C., Vella, A.: Graph-like continua, augmenting arcs, and Menger’s theorem. Combinatorica 28(5), 595–623 (2008). https://doi.org/10.1007/s00493-008-2342-9. MR2501250
We would like to thank Nathan Bowler for greatly simplifying our proof of Theorem 4.6 by pointing out the redundancy of the extra condition in the definition of pseudo-arc (cf. Lemma 4.3).
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The first author was supported by the Institute for Basic Science (IBS-R029-C1).
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Gollin, J.P., Kneip, J. Representations of Infinite Tree Sets. Order 38, 79–96 (2021). https://doi.org/10.1007/s11083-020-09529-0
- Tree sets
- Tree-like spaces
- Infinite graphs