The Classification of Homogeneous Finite-Dimensional Permutation Structures

Braunfeld, Samuel; Simon, Pierre

doi:10.37236/8321

Cited by 6 publications

(9 citation statements)

References 8 publications

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“…The proof of this last theorem requires only a small part of the paper, namely Sections 2, 3 and 7. The imprimitive case is classified in [BS18], joint with Samuel Braunfeld.…”

Section: Is Distal Of Finite Dl-dimension;mentioning

confidence: 99%

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Examples of Nip Structures

Simon¹

2015

A Guide to NIP Theories

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We classify primitive, rank 1, ω-categorical structures having polynomially many types over finite sets. For a fixed number of 4-types, we show that there are only finitely many such structures and that all are built out of finitely many linear or circular orders interacting in a restricted number of ways. As an example of application, we deduce the classification of primitive structures homogeneous in a language consisting of n linear orders as well as all reducts of such structures. * Partially supported by NSF (grant no. 1665491) and a Sloan fellowship. PreliminariesAssumption: Throughout this paper, we work in an ω-categorical structure M. That assumption will in general not be recalled, and is implicitely assumed in all statements. Linear orders and their reductsThere is only one countable homogeneous linear order: (Q, ≤). It is also the only ω-categorical primitive linear order. Its reducts follow from Cameron's result on highly homogeneous permutations groups [Cam76]: there are five of them. Apart from the trivial reduct to pure equality, there are three unstable proper reducts:• the generic betweenness relation (Q; B(x, y, z)• the generic circular order (Q; C(x, y, z)1. rk(a/D) = 0 if and only if a ∈ acl(D).2. rk(D 1 ∪ D 2 ) = max(rk(D 1 ), rk(D 2 )).3. If B 1 ⊆ B 2 , then rk(a/B 1 ) ≥ rk(a/B 2 ) 4. If D is definable over B, then there is a ∈ D such that rk(a/B) = rk(D). 5.We have rk(a/b) ≥ n + 1 if and only if there are a ′ , c ∈ M eq with a ′ ∈ acl eq (abc) \ acl eq (ac) and rk(a/a ′ bc) ≥ n.

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“…The proof of this last theorem requires only a small part of the paper, namely Sections 2, 3 and 7. The imprimitive case is classified in [BS18], joint with Samuel Braunfeld.…”

Section: Is Distal Of Finite Dl-dimension;mentioning

confidence: 99%

“…The classification of imprimitive homogeneous multi-orders is carried out in [BS18], making further use of techniques from this paper.…”

Section: Reductsmentioning

confidence: 99%

Examples of Nip Structures

Simon¹

2015

A Guide to NIP Theories

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“…The subject has important connections to permutation group theory, model theory, combinatorial enumeration, Ramsey theory, topological dynamics, and constraint satisfaction. In certain specified binary contexts, there are classification theorems of the homogeneous structures: for example for partial orders [36], coloured partial orders [38], graphs [29], digraphs [13], 'finite-dimensional permutation structures' (structures in a language with finitely many total order symbols) [6], and metrically homogeneous graphs of diameter 3 [3]. However, there is currently very little (beyond [2]) in the way of classification theorems for homogeneous structures where relation symbols have arity greater than two, and evidence from binary classifications suggests such results will be very difficult.…”

Section: Introduction 1backgroundmentioning

confidence: 99%

Set-homogeneous hypergraphs

Assari¹,

Hosseinzadeh²,

Macpherson³

2022

Preprint

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A k-uniform hypergraph M is set-homogeneous if it is countable (possibly finite) and whenever two finite induced subhypergraphs U, V are isomorphic there is g ∈ Aut(M ) with U g = V ; the hypergraph M is said to be homogeneous if in addition every isomorphism between finite induced subhypergraphs extends to an automorphism. We give four examples of countably infinite set-homogeneous k-uniform hypergraphs which are not homogeneous (two with k = 3, one with k = 4, and one with k = 6). Evidence is also given that these may be the only ones, up to complementation. For example, for k = 3 there is just one countably infinite k-uniform hypergraph whose automorphism group is not 2-transitive, and there is none for k = 4. We also give an example of a finite set-homogeneous 3-uniform hypergraph which is not homogeneous.

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“…More recently, the particular case of homogeneous structures for a language consisting of a finite set of linear orders has been completely settled by the application of ideas of the type associated with neo-stability theory (NIP) in model theory [Sim18,BrS18]. This is a very interesting case in which the application of direct amalgamation theoretic arguments is adequate in the case of at most three linear orders, but when approached in this fashion the natural analysis appears to blow up with the size the language.…”

Section: Introduction and Outlinementioning

confidence: 99%