Metrically homogeneous graphs of diameter 3

Amato, Daniela; Cherlin, Gregory; Macpherson, Dugald

doi:10.1142/s0219061320500208

Cited by 7 publications

(30 citation statements)

References 40 publications

(30 reference statements)

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“…We shall see that the most natural setting for our proof is to work with the class of all finite integer-valued antipodal metric spaces of diameter 3. Following [ACM16] we call a metric space an integer-valued metric space of diameter 3 if the distance of every two distinct points is 1, 2 or 3. It is antipodal if (1) it contains no triangle with distances 2, 2, 3, and (2) the edges with label 3 form a perfect matching (in other words, for every vertex there is precisely one antipodal vertex at distance 3).…”

Section: Introductionmentioning

confidence: 99%

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EPPA for two-graphs and antipodal metric spaces

Evans

Hubička

Konečný³

et al. 2020

Proc. Amer. Math. Soc.

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We prove that the class of finite two-graphs has the extension property for partial automorphisms (EPPA, or Hrushovski property), thereby answering a question of Macpherson. In other words, we show that the class of graphs has the extension property for switching automorphisms. We present a short, self-contained, purely combinatorial proof which also proves EPPA for the class of integer valued antipodal metric spaces of diameter 3, answering a question of Aranda et al.The class of two-graphs is an important new example which behaves differently from all the other known classes with EPPA: Two-graphs do not have the amalgamation property with automorphisms (APA), their Ramsey expansion has to add a graph, it is not known if they have coherent EPPA and even EPPA itself cannot be proved using the Herwig-Lascar theorem.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

EPPA for two-graphs and antipodal metric spaces

Evans

Hubička

Konečný³

et al. 2020

Proc. Amer. Math. Soc.

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“…Generalising the concept of distance transitivity, we say that a (countable) connected graph G is metrically homogeneous if every partial distancepreserving map of G with finite domain extends to an automorphism of G (so it is, in a sense, an EPPA-witness for itself). Cherlin [6] gave a list of countable metrically homogeneous graphs (which is conjectured to be complete and is provably complete in some cases [1,7]) in terms of classes of finite metric spaces which embed into the path-metric space of the given metrically homogeneous graph. EPPA and other combinatorial properties of classes from Cherlin's list were studied by Aranda, Bradley-Williams, Hng, Hubička, Karamanlis, Kompatscher, Pawliuk and the author [2,3,4] (see also [24]) and in [4] almost all the questions were settled, only EPPA for antipodal classes of odd diameter and bipartite antipodal classes of even diameter (see Section 2.3) remained open.…”

Section: Introductionmentioning

confidence: 99%

Extending partial isometries of antipodal graphs

Konečný

2020

Discrete Mathematics

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We prove EPPA (extension property for partial automorphisms) for all antipodal classes from Cherlin's list of metrically homogeneous graphs, thereby answering a question of Aranda et al. This paper should be seen as the first application of a new general method for proving EPPA which can bypass the lack of automorphism-preserving completions. It is done by combining the recent strengthening of the Herwig-Lascar theorem by Hubička, Nešetřil and the author with the ideas of the proof of EPPA for two-graphs by Evans et al.

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“…A metric space is homogeneous if every isomorphism, or isometry, between finite subspaces extends to an automorphism of the whole metric space.) In a recent monograph, Cherlin [Che17,Che11] gives a catalogue of metrically homogeneous graphs which is conjectured to be complete and confirmed up to diameter three [ACM16]. This is so far the most elaborate addition to the classification programme of homogeneous structures.…”

Section: Introductionmentioning

confidence: 99%

Forbidden cycles in metrically homogeneous graphs

Hubička¹,

Kompatscher²,

Konečný³

2018

Preprint

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Aranda, Bradley-Williams, Hubička, Karamanlis, Kompatscher, Konečný and Pawliuk recently proved that for every primitive 3-constrained space Γ of finite diameter δ from Cherlin's catalogue of metrically homogeneous graphs there is a finite family F of {1, 2, . . . , δ}-edge-labelled cycles such that each {1, 2, . . . , δ}-edge-labelled graph is a (not necessarily induced) subgraph of Γ if and only if it contains no homomorphic images of cycles from F . This analysis is a key to showing that the ages of metrically homogeneous graphs have Ramsey expansions and the extension property for partial automorphisms.In this paper we give an explicit description of the cycles in families F . This has further applications, for example, interpreting the graphs as semigroupvalued metric spaces or homogenizations of ω-categorical {1, δ}-edge-labelled graphs.

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Metrically homogeneous graphs of diameter 3

Abstract: We classify countable metrically homogeneous graphs of diameter 3.

Cited by 7 publications

References 40 publications

EPPA for two-graphs and antipodal metric spaces

EPPA for two-graphs and antipodal metric spaces

Extending partial isometries of antipodal graphs

Forbidden cycles in metrically homogeneous graphs

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