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Set-homogeneous structures

Banff International Research Station for Mathematical Innovation and Discovery

A countable relational structure $M$ is called $ extit{set-homogeneous}$ if whenever two finite substructures $U$, $V$ of $M$ are isomorphic, there is an automorphism of $M$ taking $U$ to $V$ (but we do not require that every isomorphism between $U$ and $V$ extends to an automorphism). This notion was originally introduced by Fraïssé, although unpublished observations had been made on it earlier by Fraïssé and Pouzet. Clearly every homogeneous structure is set-homogeneous. It is also not too difficult to construct examples of structures that are set-homogeneous but not homogeneous. It is natural to investigate the extent to which homogeneity is stronger than set-homogeneity, and this question has received some attention in the literature. For instance, it was shown by Ronse cite{Ronse1978} that any finite set-homogeneous graph is in fact homogeneous. In this talk I will give a survey of some of the known results in this area, including results on countably infinite set-homogeneous graphs due to Droste, Giraudet, Macpherson and Sauer cite{dgms}, and results on set-homogeneous directed graphs obtained in recent joint work with Macpherson, Praeger and Royle cite{gmpr}. I will also present a number of interesting conjectures and open problems that remain about set-homogeneous structures.

Mathematics; Combinatorics; Dynamical systems and ergodic theory; Logic and foundations

video/mp4

Author affiliation: University of East Anglia

2016-05-13

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/58135

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/