Some Fra ss e Classes of Finite Integral Cherlin Metric Spaces - PowerPoint PPT Presentation

Some Fra¨ ıss´ e Classes of Finite Integral Metric Spaces Gregory Some Fra¨ ıss´ e Classes of Finite Integral Cherlin Metric Spaces Metrically Ho- mogeneous Graphs Finite Distance Transitive Graphs Homogeneous Gregory Cherlin Graphs Homogeneous Metric Spaces A Catalog Special Cases Generic Cases Proofs Conclusion Bertinoro, May 27

Some Fra¨ ıss´ e Classes of Finite Integral Metric Spaces Metrically Homogeneous Graphs 1 Gregory Finite Distance Transitive Graphs Cherlin Homogeneous Graphs Metrically Ho- Homogeneous Metric Spaces mogeneous Graphs Finite Distance Transitive Graphs Homogeneous A Catalog 2 Graphs Homogeneous Special Cases Metric Spaces A Catalog Generic Cases Special Cases Proofs Generic Cases Proofs Conclusion Conclusion 3

The Classification Problem Some Fra¨ ıss´ e Classes of Finite Integral Metric Spaces Gregory Cherlin Γ connected, with graph metric d . Metrically Ho- Γ is metrically homogeneous if the metric space (Γ , d ) is mogeneous Graphs (ultra)homogeneous. Finite Distance Transitive Graphs (Cameron 1998) Classify the countable metrically Homogeneous Graphs Homogeneous homogeneous graphs. Metric Spaces Contexts: infinite distance transitive graphs, homogeneous A Catalog Special Cases graphs, homogeneous metric spaces Generic Cases Proofs Conclusion

Finite Distance Transitive Graphs Some Fra¨ ıss´ e Classes of Finite Integral Metric Spaces Gregory Cherlin distance transitivity = metric homogeneity for pairs Metrically Ho- mogeneous Graphs Smith’s Theorem: Finite Distance Transitive Graphs • Imprimitive case: Bipartite or Antipodal (or a cycle) Homogeneous Graphs Antipodal: maximal distance δ Homogeneous Metric Spaces • Reduction to the primitive case (halving, folding) A Catalog Special Cases Generic Cases Proofs Conclusion

Classification of Homogeneous Graphs Metrically homogeneous diameter ≤ 2 = Homogeneous. Some Fra¨ ıss´ e Classes of (The metric is the graph) Finite Integral Metric Spaces e Constructions: Henson graphs H n , H c Fra¨ ıss´ Gregory n Cherlin Lachlan-Woodrow 1980 The homogeneous graphs are Metrically Ho- m · K n and its complement; mogeneous Graphs The pentagon and the line graph of K 3 , 3 (3 × 3 grid) Finite Distance Transitive Graphs The Henson graphs and their complements (including Homogeneous Graphs Homogeneous the Rado graph) Metric Spaces A Catalog Method: Induction on Amalgamation Classes Special Cases Generic Cases Claim: If A is an amalgamation class of finite graphs Proofs containing all graphs of order 3, I ∞ , and K n , then A contains Conclusion every K n + 1 -free graph. Proof by induction on the order | A | where A is K n + 1 -free This doesn’t work directly, but a stronger statement can be proved by induction.

Induction via Amalgamation Some Fra¨ ıss´ e Classes of Finite Integral Metric Spaces A ′ is the set of finite graphs G such that any 1-point Gregory extension of G lies in A . Cherlin Inductive claim: Every finite graph belongs to A ′ . Metrically Ho- mogeneous Not making much progress yet, but . . . Graphs Finite Distance Transitive Graphs 1-complete: complete. 0-complete: co-complete. Homogeneous Graphs A p is the set of finite graphs G such that any finite Homogeneous Metric Spaces p -complete graph extension of G belongs to A . A Catalog A p ⊆ A ′ Special Cases Generic Cases A p is an amalgamation class Proofs Conclusion Target: The generators of A all lie in one A p , for some p .

Lachlan’s Ramsey Argument Some Fra¨ ıss´ e Classes of Finite Integral Metric Spaces How to get into A p : Gregory Cherlin Metrically Ho- 1-point extensions of a large direct sum ⊕ A i mogeneous Graphs = ⇒ Finite Distance Transitive Graphs p -extensions of one of the A i . Homogeneous Graphs Homogeneous Metric Spaces If A i is itself a direct sum of generators, we get a fixed value A Catalog Special Cases of p . Generic Cases Proofs First used for tournaments: Lachlan 1984, cf. Cherlin 1988 Conclusion

Homogeneous Metric Spaces Some Fra¨ ıss´ e Classes of Finite Integral Metric Spaces Gregory Cherlin Rational-valued Urysohn space. Z -valued Urysohn space is a metrically homogeneous Metrically Ho- mogeneous space. Graphs Finite Distance Or Z ∩ [ 0 , δ ] -valued. Transitive Graphs Homogeneous S -valued: Van Th´ e AMS Memoir 2010 Graphs Homogeneous Metric Spaces A metrically homogeneous graph of diameter δ is: A Catalog A Z -valued homogeneous metric space with bound δ , and Special Cases Generic Cases all triangles ( 1 , i , i + 1 ) allowed (connectivity). Proofs Conclusion

Some Fra¨ ıss´ e Classes of Finite Integral Metric Spaces Metrically Homogeneous Graphs 1 Gregory Finite Distance Transitive Graphs Cherlin Homogeneous Graphs Metrically Ho- Homogeneous Metric Spaces mogeneous Graphs Finite Distance Transitive Graphs Homogeneous A Catalog 2 Graphs Homogeneous Special Cases Metric Spaces A Catalog Generic Cases Special Cases Proofs Generic Cases Proofs Conclusion Conclusion 3

Special Cases Some Fra¨ ıss´ e Classes of Finite Integral Metric Spaces Gregory Cherlin Metrically Ho- Diameter ≤ 2 (Lachlan/Woodrow 1980) mogeneous Graphs Locally finite (Cameron, Macpherson) Finite Distance Transitive Graphs Homogeneous Γ 1 -exceptional Graphs Homogeneous Metric Spaces Imprimitive (Smith’s Theorem) A Catalog Special Cases Generic Cases Proofs Conclusion

The Locally Finite Case Some Fra¨ ıss´ e Finite of diameter at least 3 and vertex degree at least 3: Classes of Antipodal double covers of certain finite homogeneous Finite Integral Metric Spaces graphs (Cameron 1980) Gregory Cherlin Metrically Ho- mogeneous Graphs Finite Distance Transitive Graphs Homogeneous Graphs Homogeneous Metric Spaces A Catalog Special Cases Generic Cases Proofs Conclusion Figure: Antipodal Double cover of C 5 Infinite, Locally Finite: Tree-like T r , s (Macpherson 1982) Construction:

The graphs T r , s Some Fra¨ ıss´ e Classes of Finite Integral Metric Spaces The trees T ( r , s ) : Alternately r -branching and s -branching. Gregory Bipartite, metrically homogeneous if the two halves of the Cherlin partition are kept fixed. Metrically Ho- mogeneous The graph obtained by “halving” on the r -branching side is Graphs Finite Distance T r , s . Transitive Graphs Homogeneous Graphs Each vertex lies at the center of a bouquet of r s -cliques. Homogeneous Metric Spaces Another point of view: the graph on the neighbors of a fixed A Catalog Special Cases vertex: Generic Cases Proofs Γ 1 : r · K s − 1 . Conclusion From this point of view, we may also take r or s to be infinite!

Γ 1 Some Fra¨ ıss´ e Classes of Γ i = Γ i ( v ) : Distance i , with the induced metric. Finite Integral Metric Spaces Remark Gregory Cherlin If distance 1 occurs, then the connected components of Γ i Metrically Ho- are metrically homogeneous. mogeneous Graphs Finite Distance In particular Γ 1 is a homogeneous graph. Transitive Graphs Homogeneous Graphs Exceptional Cases: finite, imprimitive, or H c n . Homogeneous Metric Spaces The finite case is Cameron+Macpherson, the imprimitive A Catalog case leads back to T r , s with r or s infinite, and H c n does not Special Cases Generic Cases occur for n > 2 (Cherlin 2011) Proofs Conclusion In other words, the nonexceptional cases are I ∞ Henson graphs H n including Rado’s graph.

Imprimitive Graphs Some Fra¨ ıss´ e “Smith’s Theorem” (Amato/Macpherson, Cherlin): Classes of Finite Integral Part I: Bipartite or antipodal, and in the antipodal case with Metric Spaces classes of order 2 and the metric antipodal law for the Gregory Cherlin pairing: d ( x , y ′ ) = δ − d ( x , y ) Metrically Ho- mogeneous Graphs Hence no triangles of diameter greater than 2 δ : Finite Distance Transitive Graphs Homogeneous Graphs d ( x , z ) ≤ d ( x , y ′ ) + d ( y ′ , z ) = 2 δ − d ( x , y ) − d ( x , z ) Homogeneous Metric Spaces A Catalog Part II: The bipartite case reduces by halving to a case in Special Cases Generic Cases which Γ 1 is the Rado graph. Proofs Conclusion On the other hand, the antipodal case does not reduce: while distance transitivity is inherited after “folding,” metric homogeneity is not. There is also a bipartite antipodal case.

Some Amalgamation Classes Some Fra¨ ıss´ e Within A δ : finite integral metric spaces with bound δ : Classes of Finite Integral Metric Spaces A δ K , even: No odd cycles below 2 K + 1. Gregory A δ Cherlin C , bounded: Perimeter at most C . Metrically Ho- ( 1 , δ ) -constraints. mogeneous Graphs The first two classes are given (implicitly) in Finite Distance Transitive Graphs Homogeneous Komjath/Mekler/Pach 1988 as examples of constraints Graphs Homogeneous admitting a universal graph, which is constructed by Metric Spaces A Catalog amalgamation. Special Cases The last is a generalization of Henson’s construction. A Generic Cases Proofs ( 1 , δ ) -space is a space in which only the distances 1 and δ Conclusion occur (a vacuous condition if δ = 2). Any set S of ( 1 , δ ) -constraints may be imposed. Mixing: A δ K , C ; S