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Finite presentation of homogeneous graphs, posets and Ramsey classes

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Abstract

It is commonly believed that one can prove Ramsey properties only for simple and “well behaved” structures. This is supported by the link of Ramsey classes of structures with homogeneous structures. We outline this correspondence in the context of the Classification Programme for Ramsey classes. As particular instances of this approach one can characterize all Ramsey classes of graphs, tournaments and partial ordered sets and also fully characterize all monotone Ramsey classes of relational systems (of any type). On the other side of this spectrum many homogeneous structures allow a concise description (called here a finite presentation) by means of all finite models of a suitable theory. Extending classical work of Rado (for the random graph) we find a finite presentation for each of the above classes where the classification problem is solved: (undirected) graphs, tournaments and partially ordered sets. The main result of the paper is a construction of classesP ∈ andP f of finite structures which are isomorphic to the generic (i.e. homogeneous and universal) partially ordered set. Somehow surprisingly, the structureP ∈ extends Conway’s surreal numbers and their linear ordering.

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Authors and Affiliations

  1. Department of Applied Mathematics and Institute of Theoretical Computer Sciences (ITI), Charles University, Malostranské nám. 25, 11800, Praha, Czech Republic

    Jan Hubička & Jaroslav Nešetřil

Authors
  1. Jan Hubička
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  2. Jaroslav Nešetřil
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Dedicated to Professor Hillel Furstenberg

Supported by Grants LN00A56 and 1M0021620808 of the Czech Ministry of Education and ICREA, Barcelona, Spain.

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Hubička, J., Nešetřil, J. Finite presentation of homogeneous graphs, posets and Ramsey classes. Isr. J. Math. 149, 21–44 (2005). https://doi.org/10.1007/BF02772535

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