Abstract
We investigate the combinatorial properties of the Fraïssé class of structures induced by the leaf sets of boron trees — graph-theoretic binary trees without an assigned root — and compute their Ramsey degrees. The Ramsey degree of a boron tree structure is shown to equal the number of its possible orientations, which are herein defined to depend on the embedding of the said structure. One direction of this computation involves an asymmetric version of the Graham-Rothschild theorem. By expanding these structures to oriented ones, we arrive at a Fraïssé class which possesses the Ramsey property. Consequently, the automorphism group of its Fraïssé limit is extremely amenable, i.e., it possesses a very strong fixed point property. Furthermore, we construct the universal minimal flow of the automorphism group of this limit.
Similar content being viewed by others
References
P. J. Cameron: Oligomorphic Permutation Groups, London Math. Society Lecture Note Series 152, 1990.
P. J. Cameron: Some treelike objects, Quart. J. Math. Oxford Ser. (2) 38(150) (1987), 155–183.
W. Deuber: A generalization of Ramsey’s theorem for regular trees, J. Combinatorial Theory Ser. B 18 (1975), 18–23.
W. L. Fouché: Symmetries and Ramsey properties of trees, Discrete Mathematics 197/198 (1999), 325–330.
R. Fraïssé: Sur l’extension aux relations de quelques propriétés des ordres (in French), Ann. Sci. Ecole Norm. Sup. 71(3) (1954), 363–388.
R. Fraïssé: Theory of Relations, Elsevier, Rev. ed., 2000.
R. L. Graham, B. L. Rothschild: Ramsey’s theorem for n-parameter sets, Trans. Amer. Math. Soc. 159 (1971), 257–292.
A. S. Kechris, V. G. Pestov, S. Todorcevic: Fraïssé limits, Ramsey theory, and topological dynamics of automorphism groups, Geom. Funct. Anal. 15(1) (2005), 106–189.
K. R. Milliken: A Ramsey theorem for trees, J. Combin. Theory Ser. A 26(3) (1979), 215–237.
J. Nešetřil: Ramsey Classes and Homogeneous Structures, Comb. Probab. Comput. 14(1) (2005), 171–189.
J. Nešetřil: Ramsey Theory, Handbook of Combinatorics, (R. Graham, et al., eds.), Elsevier (1995), 1331–1403.
J. Nešetřil, V. Rödl: The partite construction and Ramsey set systems, Discrete Mathematics 75(1–3) (1989), 327–334.
V. Pestov: Dynamics of Infinite-dimensional Groups: The Ramsey-Dvoretzky-Milman Phenomenon, American Math. Society University Lecture Series 40, 2006.
B. Voigt: The partition problem for finite abelian groups, J. Combin. Theory A 28 (1980), 257–271.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Jasiński, J. Ramsey degrees of boron tree structures. Combinatorica 33, 23–44 (2013). https://doi.org/10.1007/s00493-013-2723-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00493-013-2723-6