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Extension Properties of Structures Charles University in Prague Faculty of Mathematics and Physics DOCTORAL THESIS David Hartman Extension properties of structures Computer Science Institute of Charles University Supervisor of the doctoral thesis: Prof. RNDr. Jaroslav Neˇsetˇril,DrSc. Study programme: Informatics Specialization: 414 Prague 2014 I would like to express my gratitude to my advisor Prof. RNDr. Jaroslav Neˇsetˇril, DrSc. for patient leadership and the inspiration he gave me to explore many areas of mathematics as well as his inspiration in personal life more generally. Further- more, I would like to thank my coauthors Jan Hubiˇcka and Dragan Maˇsulovi´c for helping me to establish ideas and for the realization of these ideas in their final published form. Moreover thanks are due to Peter J. Cameron for posing several questions about classes of homomorphism-homogeneous structures in our personal communication that subsequently led to one of the publications as well as to John K. Truss whose interest in my work after the first Workshop on Ho- mogeneous Structures supported my further progress. Many thanks belong to Andrew Goodall who provide an extensive feedback for many parts of the thesis. Finally, the greatest thanks belong to my family, my beloved wife Mireˇcka and our children Emma Alexandra and Hugo, as altogether the main source of joy, happiness and in fact meaning of life. I declare that I carried out this doctoral thesis independently, and only with the cited sources, literature and other professional sources. I understand that my work relates to the rights and obligations under the Act No. 121/2000 Coll., the Copyright Act, as amended, in particular the fact that the Charles University in Prague has the right to conclude a license agreement on the use of this work as a school work pursuant to Section 60 paragraph 1 of the Copyright Act. In ........ date ............ signature of the author N´azevpr´ace: Rozˇsiˇruj´ıc´ıvlastnosti struktur Autor: David Hartman Katedra: Informatick´y´ustav Univerzity Karlovy Vedouc´ıdisertaˇcn´ıpr´ace: Prof. RNDr. Jaroslav Neˇsetˇril,DrSc. Abstrakt: Tato pr´acerozeb´ır´avlastnost relaˇcn´ıch struktur, kter´aimplikuje je- jich vysokou symetriˇcnost. Strukturu nazveme homogenn´ıpokud lze libovoln´e lok´aln´ızobrazen´ırozˇs´ıˇritna zobrazen´ınad celou strukturou a to pro libovolnou volbu koneˇcn´evzorov´emnoˇziny. Typ lok´aln´ıhoa glob´aln´ıhozobrazen´ıpotom urˇcujer˚uzn´etypy homogenity. Prominentn´ım´ıstom´aultrahomogenita, kter´a oznaˇcujestrukturu, pro kterou libovoln´ylok´aln´ı isomorfismus nad koneˇcn´ymi podstrukturami je rozˇsiˇriteln´yna automorfismus. Na rozd´ılod graf˚uje klasi- fikace ultrahomogenn´ıch relaˇcn´ıch struktur st´ale otevˇren´ymprobl´emem. C´ılem pr´aceje charakterizovat “vzd´alenost"od homogenity a to dvˇemazp˚usoby. Ne- jprve zvyˇsuje\sloˇzitoststruktury" pˇrid´av´an´ımrelac´ıa sleduje zmˇeny klasifikace homogenn´ıch struktur. To vede k nˇekolika klasifikac´ımhomomorfnˇe-homogenn´ıch L-obarviteln´ych graf˚upro r˚uzn´e L, kde L-obarviteln´ygraf je graf, kde vrcholy a hrany dost´avaj´ımnoˇziny barev z ˇc´asteˇcnˇeuspoˇr´adan´emnoˇziny L. Na to navazuj´ı v´ysledkya diskuze nad hierarchi´ıtˇr´ıddefinovanou skrze r˚uzn´etypy homogenity s ohledem na koincidenci jednotliv´ych tˇr´ıd.Druh´ypohled zkoum´apro dan´estruk- tury jak minim´alnˇerozˇs´ıˇritjejich jazyk, abychom dos´ahlihomogenity. V´ysledky se t´ykaj´ı relaˇcn´ı komplexity koneˇcn´ych graf˚ua d´alejej´ı meze pro nekoneˇcnˇe spoˇcetn´erelaˇcn´ıstruktury definovan´etˇr´ıdouzak´azan´ych homomorfism˚u. Kl´ıˇcov´aslova: relaˇcn´ı struktury, ultrahomogen´ı struktury, homomorfnˇeho- mogenn´ıstruktury, obarven´egrafy, relaˇcn´ısloˇzitost,komplexn´ıs´ıtˇe Title: Extension property of structures Author: David Hartman Department: Computer Science Institute of Charles University Supervisor: Prof. RNDr. Jaroslav Neˇsetˇril,DrSc. Abstract: This work analyses properties of relational structures that imply a high degree of symmetry. A structure is called homogeneous if every mapping from any finite substructure can be extended to a mapping over the whole struc- ture. The various types of these mappings determine corresponding types of homogeneity. A prominent position belongs to ultrahomogeneity, for which every local isomorphism can be extended to an automorphism. In contrast to graphs, the classification of ultrahomogeneous relational structures is still an open prob- lem. The task of this work is to characterize \the distance" to homogeneity using two approaches. Firstly, the classification of homogeneous structures is studied when the \complexity" of a structure is increased by introducing more relations. This leads to various classifications of homomorphism-homogeneous L-colored graphs for different L, where L-colored graphs are graphs having sets of colors from a partially ordered set L assigned to vertices and edges. Moreover a hier- archy of classes of homogeneous structures defined via types of homogeneity is studied from the viewpoint of classes coincidence. The second approach analy- ses for fixed classes of structures the least way to extend their language so as to achieve homogeneity. We obtain results about relational complexity for finite graphs and bounds on this complexity for countably infinite structures defined via classes of forbidden homomorphisms. Keywords: relational structures, ultrahomogeneous structures, homomorphism- homogeneous structures, colored graphs, relational complexity, complex networks Contents 1 Introduction 3 1.1 Symmetry of structures . .4 1.2 Ultrahomogeneity . .5 1.2.1 The Rado graph . .5 1.2.2 The Fra¨ıss´emethod . .8 1.3 Combinatorial definitions . 12 1.3.1 Graph-theoretic definitions . 12 1.3.2 Partially ordered sets . 14 1.3.3 Digraphs . 14 1.3.4 Multicolored graphs . 15 1.4 Classification results for homogeneity . 16 1.5 Structures near homogeneity . 18 1.5.1 Models and theories . 18 1.5.2 !-categorical structures . 20 1.6 Class of forbidden structures . 20 1.7 Homomorphism homogeneity . 21 1.7.1 Homomorphism-homogeneous graphs . 21 1.7.2 Fra¨ıss´eanalogue for homomorphism-homogeneity . 23 1.8 Classification results for homomorphism homogeneity . 25 1.9 Other variants of homogeneity . 27 1.10 Summary . 29 2 Complex network symmetries 31 2.1 Network construction . 34 2.2 Characteristics of complex networks . 35 2.3 The brain network . 38 2.3.1 Defining vertices . 38 2.3.2 Defining edges . 39 2.3.3 Linear surrogate datasets . 42 2.3.4 Testing of nonlinearity influence . 42 2.4 The climate network . 46 2.4.1 Climatic data . 47 2.4.2 Connectivity in climate networks . 48 2.4.3 Directed complex networks for climate . 49 2.5 Small-world in randomly connected systems . 51 2.6 Concluding remarks and motivation . 54 3 Bicolored graphs 58 3.1 Bigraphs with a red-blue edge . 58 3.2 Bigraphs without a red-blue edge . 60 3.3 Concluding remarks . 62 1 4 L-colored graphs 65 4.1 MH-homogeneous L-colored graph . 65 4.2 L-colored graphs over chains . 68 4.3 L-colored graphs over diamonds . 70 4.4 Concluding remarks . 73 5 Morphism extension classes 76 5.1 Known hierarchies . 76 5.2 Classes for L-colored graphs . 78 5.3 Concluding remarks . 80 6 Relational complexity 81 6.1 Introduction . 82 6.2 Complexity of relational structures . 84 6.3 Basic properties . 84 6.3.1 Graphs of complexity 1 . 86 6.3.2 Graphs of complexity 2 . 88 6.4 Finite graphs with large complexity . 89 6.5 Complexity of infinite structures . 91 6.5.1 Upper bounds on relational complexity . 93 6.5.2 Lower bounds on relational complexity . 96 6.6 Concluding remarks . 97 7 Conclusion 100 7.1 Homogeneous structures as motivation for complex networks . 101 7.2 Colored graphs . 102 7.3 Morphism extension classes . 102 7.4 Relational complexity . 103 References 106 List of Abbreviations 117 Glossary 120 2 1. Introduction The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms of the beautiful. Aristotle There is a long-standing effort, albeit not precisely defined, to discover and explore beauty in mathematical objects of all kinds. The roots of beauty are often found to be connected with symmetries of the corresponding object. There are many ways how symmetry can be defined, usually through different representa- tions of the object under study. Using notions from model theory, an interesting type of symmetry { and from a specific point of view one of the most fundamen- tal { is achieved when mathematical objects are studied from the perspective of \structure-preserving" extensions of morphisms over the structure itself. To be able to define this kind of symmetry in a more formal way, let us first define the necessary notions that are required. i A relational structure A is a pair (A; RA), where RA is a tuple (RA : i I) i δi i 2 of relations such that RA A (i.e. RA is a δi-ary relation on A). It should be mentioned that constant⊆ and function symbols can also be defined for general structures, but we do not consider these here { for more information see [67]. The underlying set A is called the domain of A, whose cardinality is the corresponding cardinality of the whole relational structure, i.e. A = A . Elements of the j j j j domain are usually denoted by a1; a2;:::, or by x1; x2;:::. A finite sequence of elements (a1; a2; : : : ; an) or (x1; x2; : : : ; xn) is called an n-tuple or simply a tuple, and denoted by a or x. The signature L of a relational structure A is a set of n-ary relational symbols given for each separate n. A signature represents a language from which the relational structure is constructed and it can be usually assumed to be read simply from the definition of the structure. When there is a need to specify the language for a particular structure we call such a structure an L-structure.
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