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Random l-colourable structures with a pregeometry
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.ORCID iD: 0000-0002-4477-4476
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2017 (English)In: Mathematical logic quarterly, ISSN 0942-5616, E-ISSN 1521-3870, Vol. 63, no 1-2, p. 32-58Article in journal (Refereed) Published
##### Abstract [en]

We study finite -colourable structures with an underlying pregeometry. The probability measure that is usedcorresponds to a process of generating such structures by which colours are first randomly assigned to all1-dimensional subspaces and then relationships are assigned in such a way that the colouring conditions aresatisfied but apart from this in a random way. We can then ask what the probability is that the resulting structure,where we now forget the specific colouring of the generating process, has a given property. With this measurewe get the following results: (1) A zero-one law. (2) The set of sentences with asymptotic probability 1 has anexplicit axiomatisation which is presented. (3) There is a formula ξ (x, y) (not directly speaking about colours)such that, with asymptotic probability 1, the relation “there is an -colouring which assigns the same colourto x and y” is defined by ξ (x, y). (4) With asymptotic probability 1, an -colourable structure has a unique-colouring (up to permutation of the colours).

##### Place, publisher, year, edition, pages
Wiley-VCH Verlagsgesellschaft, 2017. Vol. 63, no 1-2, p. 32-58
##### National Category
Algebra and Logic
##### Research subject
Mathematical Logic
##### Identifiers
ISI: 000400361900003OAI: oai:DiVA.org:uu-321515DiVA, id: diva2:1093452
Available from: 2017-05-06 Created: 2017-05-06 Last updated: 2017-11-28Bibliographically approved
##### In thesis
1. Limit Laws, Homogenizable Structures and Their Connections
Open this publication in new window or tab >>Limit Laws, Homogenizable Structures and Their Connections
2018 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Alternative title[sv]
Gränsvärdeslagar, Homogeniserbara Strukturer och Deras Samband
##### Abstract [en]

This thesis is in the field of mathematical logic and especially model theory. The thesis contain six papers where the common theme is the Rado graph R. Some of the interesting abstract properties of R are that it is simple, homogeneous (and thus countably categorical), has SU-rank 1 and trivial dependence. The Rado graph is possible to generate in a probabilistic way. If we let K be the set of all finite graphs then we obtain R as the structure which satisfy all properties which hold with assymptotic probability 1 in K. On the other hand, since the Rado graph is homogeneous, it is also possible to generate it as a Fraïssé-limit of its age.

Paper I studies the binary structures which are simple, countably categorical, with SU-rank 1 and trivial algebraic closure. The main theorem shows that these structures are all possible to generate using a similar probabilistic method which is used to generate the Rado graph. Paper II looks at the simple homogeneous structures in general and give certain technical results on the subsets of SU-rank 1.

Paper III considers the set K consisting of all colourable structures with a definable pregeometry and shows that there is a 0-1 law and almost surely a unique definable colouring. When generating the Rado graph we almost surely have only rigid structures in K. Paper IV studies what happens if the structures in K are only the non-rigid finite structures. We deduce that the limit structures essentially try to stay as rigid as possible, given the restriction, and that we in general get a limit law but not a 0-1 law.

Paper V looks at the Rado graph's close cousin the random t-partite graph and notices that this structure is not homogeneous but almost homogeneous. Rather we may just add a definable binary predicate, which hold for any two elemenets which are in the same part, in order to make it homogeneous. This property is called being homogenizable and in Paper V we do a general study of homogenizable structures. Paper VI conducts a special case study of the homogenizable graphs which are the closest to being homogeneous, providing an explicit classification of these graphs.

##### Place, publisher, year, edition, pages
Uppsala: Department of Mathematics, 2018. p. 43
##### Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 104
##### Keywords
Model theory, random structure, finite model theory, simple theory, homogeneous structure, countably categorical, 0-1 law
##### National Category
Algebra and Logic
##### Research subject
Mathematical Logic; Mathematics
##### Identifiers
urn:nbn:se:uu:diva-330142 (URN)978-91-506-2672-8 (ISBN)
##### Public defence
2018-02-16, Polhemssalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:15 (English)
##### Supervisors
Available from: 2018-01-17 Created: 2017-11-28 Last updated: 2018-02-09

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#### Authority records BETA

Ahlman, OveKoponen, Vera

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