Classifying Homoge- neous Structures Gregory Classifying Homogeneous Structures Cherlin Introduction The finite case Gregory Cherlin Directed Graphs Homogeneous Ordered Graphs Graphs as Metric Spaces November 27 Banff
Classifying Homoge- neous Structures Introduction 1 Gregory Cherlin Introduction The finite case 2 The finite case Directed Graphs Directed Graphs 3 Homogeneous Ordered Graphs Homogeneous Ordered Graphs 4 Graphs as Metric Spaces Graphs as Metric Spaces 5
The first classification theorem Classifying Homoge- neous Structures Theorem (Fraïssé’s Classification Theorem) Gregory Cherlin Countable homogeneous structures correspond to Introduction amalgamation classes of finite structures. The finite case The Fraïssé limit: Q = lim L ( L : finite linear orders). Directed Graphs Homogeneous Ordered Graphs Graphs as Metric Spaces
The first classification theorem Classifying Homoge- neous Structures Theorem (Fraïssé’s Classification Theorem) Gregory Cherlin Countable homogeneous structures correspond to Introduction amalgamation classes of finite structures. The finite case The Fraïssé limit: Q = lim L ( L : finite linear orders). Directed Graphs Homogeneous random graph, generic triangle-free graph Ordered Graphs Graphs as Metric Spaces
The first classification theorem Classifying Homoge- neous Structures Theorem (Fraïssé’s Classification Theorem) Gregory Cherlin Countable homogeneous structures correspond to Introduction amalgamation classes of finite structures. The finite case The Fraïssé limit: Q = lim L ( L : finite linear orders). Directed Graphs Homogeneous random graph, generic triangle-free graph Ordered Graphs Graphs as Good for existence: is it also good for non-existence? Metric Spaces (classification). “Short answers to simple questions:” Yes. Longer answer: sometimes . . .
In more detail: Classifying Homoge- neous General theory for the finite case (Lachlan; CSFG Structures meets model theory) Gregory Cherlin A few cases of combinatorial interest fully classified, or Introduction conjectured The finite case Some sporadics, and some families, identified via Directed classification Graphs Homogeneous Cases of particular interest: Ramsey classes Ordered Graphs Graphs as Metric Spaces
In more detail: Classifying Homoge- neous General theory for the finite case (Lachlan; CSFG Structures meets model theory) Gregory Cherlin A few cases of combinatorial interest fully classified, or Introduction conjectured The finite case Some sporadics, and some families, identified via Directed classification Graphs Homogeneous Cases of particular interest: Ramsey classes Ordered Graphs Graphs as After a glance at the finite case, I will discuss three cases I Metric Spaces have been involved with (2 of them lately): directed graphs; ordered graphs; graphs as metric spaces The key: Lachlan’s classification of the homogeneous tournaments.
Classifying Homoge- neous Structures Introduction 1 Gregory Cherlin Introduction The finite case 2 The finite case Directed Graphs Directed Graphs 3 Homogeneous Ordered Graphs Homogeneous Ordered Graphs 4 Graphs as Metric Spaces Graphs as Metric Spaces 5
Finite homogeneous graphs Classifying Homoge- neous Structures Gregory Cherlin Sheehan 1975, Gardiner 1976 Introduction The finite case C 5 , E ( K 3 , 3 ) , m · K n Directed Graphs Homogeneous Ordered Graphs Graphs as Metric Spaces
Finite homogeneous graphs Classifying Homoge- neous Structures Gregory Cherlin Sheehan 1975, Gardiner 1976 Introduction The finite case C 5 , E ( K 3 , 3 ) , m · K n Directed Graphs Homogeneous Lachlan’s view: two sporadics and a set of approximations Ordered Graphs to ∞ · K ∞ . Graphs as Metric Spaces
Lachlan’s Finiteness Theorem Classifying Homoge- neous Structures Given a finite relational language, there are finitely many Gregory homogeneous structures Γ i such that Cherlin The finite homogeneous structures are the Introduction homogeneous substructures of the Γ i . The finite case Directed The (model-theoretically) stable homogenous Graphs structures are the homogeneous substructures. Homogeneous Ordered Graphs Graphs as Metric Spaces
Lachlan’s Finiteness Theorem Given a finite relational language, there are finitely many Classifying Homoge- homogeneous structures Γ i such that neous Structures The finite homogeneous structures are the Gregory Cherlin homogeneous substructures of the Γ i . The (model-theoretically) stable homogenous Introduction The finite case structures are the homogeneous substructures. Directed Graphs Corollary: a stable homogeneous structure can be Homogeneous approximated by finite homogeneous structures. Ordered Graphs This is not true for the random graph—which can be Graphs as approximated by finite structures, but not by finite Metric Spaces homogeneous ones.
Lachlan’s Finiteness Theorem Classifying Homoge- neous Structures Given a finite relational language, there are finitely many Gregory homogeneous structures Γ i such that Cherlin The finite homogeneous structures are the Introduction homogeneous substructures of the Γ i . The finite case Directed The (model-theoretically) stable homogenous Graphs structures are the homogeneous substructures. Homogeneous Ordered Graphs Graphs as Division of Labour Metric Spaces Group theory: primitive structures Model theory: imprimitive structures (modulo primitive)
Binarity Conjecture What if we bound the relational complexity of the language, Classifying Homoge- but not the number of relations? neous Structures Gregory Cherlin Introduction The finite case Directed Graphs Homogeneous Ordered Graphs Graphs as Metric Spaces Halford et al., Psychological Science, (2005)
Binarity Conjecture Classifying Conjecture Homoge- neous Structures A finite primitive homogeneous binary structure is Gregory Equality on n points; or Cherlin An oriented p-cycle; or Introduction The finite case An affine space over a finite field, equipped with an Directed anisotropic quadratic form. Graphs Homogeneous Ordered Graphs Graphs as Metric Spaces
Binarity Conjecture Classifying Conjecture Homoge- neous Structures A finite primitive homogeneous binary structure is Gregory Equality on n points; or Cherlin An oriented p-cycle; or Introduction The finite case An affine space over a finite field, equipped with an Directed anisotropic quadratic form. Graphs Homogeneous Ordered Case Division. Graphs Affine Non-affine Graphs as Metric Spaces (abelian normal sub- (none) group) Known Reduced to almost simple case (Wiscons)
Classifying Homoge- neous Structures Introduction 1 Gregory Cherlin Introduction The finite case 2 The finite case Directed Graphs Directed Graphs 3 Homogeneous Ordered Graphs Homogeneous Ordered Graphs 4 Graphs as Metric Spaces Graphs as Metric Spaces 5
Development Classifying Existence Henson 1973 (2 ℵ 0 ) Homoge- neous Structures Partially ordered sets Schmerl 1979 Gregory Graphs Lachlan-Woodrow 1980 (induction on Cherlin amalgamation classes) Introduction Tournaments Lachlan 1984 (Ramsey argument) The finite case Directed Digraphs Cherlin 1998 (L/H Smackdown) Graphs Homogeneous Ordered Graphs Graphs as Metric Spaces
Development Classifying Existence Henson 1973 (2 ℵ 0 ) Homoge- neous Structures Partially ordered sets Schmerl 1979 Gregory Graphs Lachlan-Woodrow 1980 (induction on Cherlin amalgamation classes) Introduction Tournaments Lachlan 1984 (Ramsey argument) The finite case Directed Digraphs Cherlin 1998 (L/H Smackdown) Graphs Homogeneous C ATALOG Ordered Graphs 1. Composite / degenerate I n [ T ] , T [ I n ] Graphs as Metric Spaces 2. Twisted imprimitive double covers, generic multipartite, semigeneric 3. Exceptional primitive S ( 3 ) , P , P ( 3 ) 4. Free amalgamation Omit I n or tournaments.
The Ramsey Classes Classifying Homoge- Ramsey precompact expansions of homogeneous neous directed graphs — Jakub Jasi´ Structures nski, Claude Laflamme, Lionel Gregory Nguyen Van Thé, Robert Woodrow Cherlin (arxiv 24 Oct 2013–23 Jul 2014 (v3)) Introduction In 2005, Kechris, Pestov and Todorcevic The finite case provided a powerful tool . . . More recently, the Directed Graphs framework was generalized allowing for further Homogeneous applications, and the purpose of this paper is to Ordered Graphs apply these new methods in the context of Graphs as homogeneous directed graphs. In this paper, we Metric Spaces show that the age of any homogeneous directed graph allows a Ramsey precompact expansion. Moreover, we . . . describe the respective universal minimal flows [for Aut (Γ) ] .
LW-L method Classifying Homoge- neous (Lachlan/Woodrow 1980, Lachlan 1984) Structures Gregory Cherlin C ATALOG : T OURNAMENTS Introduction I 1 , Q Orders The finite case C 3 , S ( 2 ) Local Orders Directed Graphs Generic T ∞ Homogeneous Ordered Graphs Graphs as Metric Spaces
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