Ramsey degrees of boron tree structures

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.

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Correspondence to Jakub Jasiński.

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Jasiński, J. Ramsey degrees of boron tree structures. Combinatorica 33, 23–44 (2013). https://doi.org/10.1007/s00493-013-2723-6

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Mathematics Subject Classification (2010)

  • 05D10
  • 05C05
  • 03C13
  • 03C15
  • 37B05
  • 43A07
  • 54H20