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On Fraïssé's conjecture for linear orders of finite Hausdorff rank

2009
*
Annals of Pure and Applied Logic
*

We then show that

doi:10.1016/j.apal.2009.01.007
fatcat:mf7iar5vbrawfkxybpxi547744
*Fraïssé's**conjecture*restricted to*linear**orders**of**finite**Hausdorff**rank*is provable in ACA + 0 + "ϕ 2 (0) is well-*ordered*" and, over RCA 0 , implies ACA ′ 0 + "ϕ 2 (0) is well-*ordered*... We prove that the maximal*order*type*of*the wqo*of**linear**orders**of**finite**Hausdorff**rank*under embeddability is ϕ 2 (0), the first fixed point*of*the ε-function. ... )*of**finite**Hausdorff**rank*. ...##
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Up to equimorphism, hyperarithmetic is recursive

2005
*
Journal of Symbolic Logic (JSL)
*

*On*the way to our main result we prove that a

*linear*

*ordering*has

*Hausdorff*

*rank*less than if and only if it is equimorphic to a recursive

*one*. ... As a corollary

*of*our proof we prove that, given a recursive ordinal α, the partial

*ordering*

*of*equimorphism types

*of*

*linear*

*orderings*

*of*

*Hausdorff*

*rank*at most α

*ordered*by embeddablity is recursively ... Since every scattered

*linear*

*ordering*is equimorphic to a

*finite*sum

*of*1s and h-indecomposable

*linear*

*orderings*,

*for*every

*linear*

*ordering*

*of*

*Hausdorff*

*rank*less than or equal to α, there is a σ ∈Â α ...

##
###
The reverse mathematics of wqos and bqos
[article]

2019
*
arXiv
*
pre-print

The classification from the reverse mathematics viewpoint

arXiv:1707.08365v5
fatcat:ym22ayu7ibhdtfrqnf2pi5ko44
*of*both kinds*of*results provides interesting challenges, and we cover also recent advances*on*some long standing open problems. ... We consider both elementary results (such as the equivalence*of*different definitions*of*the concepts, and basic closure properties) and more advanced theorems. ...*On*the other hand, Marcone and Montalbán [MM09] studied the restriction*of**Fraïssé's**conjecture*to*linear**orders**of**finite**Hausdorff**rank*. ...##
###
Laver and set theory

2016
*
Archive for Mathematical Logic
*

In this commemorative article, the work

doi:10.1007/s00153-015-0462-7
fatcat:5ziwvh27arbprhsdlmku4f2npe
*of*Richard Laver is surveyed in its full range and extent. ... Doctoral students*of*Richard Laver Stephen Grantham, An analysis*of*Galvin In addition to having these doctoral students at Boulder, Laver was*on*the thesis committees*of*, among many: ...*Fraïssé's**conjecture*Laver [54, 55] in his doctoral work famously established*Fraïssé's**Conjecture*, a basicsounding statement about countable*linear**orderings*that turned out to require a substantial ...##
###
Countably Complementable Linear Orderings

2007
*
Order
*

Using similar methods and introducing the notion

doi:10.1007/s11083-006-9049-6
fatcat:eur6hi7wjbg2ljebi3uamryyhe
*of*weakly countably complementable*linear**orderings*, we answer a question posed by Rosenstein and prove the countable case*of*a*conjecture**of*Hagendorf, ... We say that a countable*linear**ordering*L is countably complementable if there exists a*linear**ordering*L, possibly uncountable, such that*for*any countable*linear**ordering*B, L does not embed into B if ...*One**of*them is the celebrated*Fraïssé's**Conjecture*, which states that in L,*ordered*by embeddability, there are no infinite strictly descending sequences and no infinite antichains. ...##
###
Open Questions in Reverse Mathematics

2011
*
Bulletin of Symbolic Logic
*

We present a list

doi:10.2178/bsl/1309952320
fatcat:maim2j6vovfcpjjkrzm7xeicqi
*of*open questions in reverse mathematics, including some relevant background information*for*each question. ... We also mention some*of*the areas*of*reverse mathematics that are starting to be developed and where interesting open question may be found. ... Marcone and Montalbán [MM09] considered L ω , the class*of**linear**ordering**of**finite**Hausdorff**rank*. ...##
###
Equimorphy -- The Case of Chains
[article]

2014
*
arXiv
*
pre-print

Such structures cannot be expected to be isomorphic, and in this paper we investigate the special case

arXiv:1407.2894v1
fatcat:r7x5ijysgvf7tchkikgqa4s7ve
*of**linear**orders*, here also called chains. ... In particular we provide structure results*for*chains having less than continuum any isomorphism classes*of*equimorphic chains. ... In particular we assume the reader to be generally familiar with the notion*of*indecomposability and*Hausdorff**rank**of*a*linear**order*although we briefly review these notions below. ...##
###
The maximal linear extension theorem in second order arithmetic

2011
*
Archive for Mathematical Logic
*

We show that the maximal

doi:10.1007/s00153-011-0231-1
fatcat:nau46smsnzczpojbwq5ik2vqqy
*linear*extension theorem*for*well partial*orders*is equivalent over RCA_0 to ATR_0. ... Analogously, the maximal chain theorem*for*well partial*orders*is equivalent to ATR_0 over RCA_0. ... In [MM09] the computation*of*the maximal*order*type*of*the scattered*linear**orders**of**finite**Hausdorff**rank*is instrumental in the reverse mathematics results about the restriction*of**Fraïssé's**conjecture*...##
###
Well Quasi-Orders in Computer Science (Dagstuhl Seminar 16031)

2016
*
Dagstuhl Reports
*

, topological complexity

doi:10.4230/dagrep.6.1.69
dblp:journals/dagstuhl-reports/Goubault-Larrecq16
fatcat:o3uwzu5ptfavfes6kgexpi4a6q
*of*computational problems*on*continuous functions). ...*of*Well quasi-*orders*(known as the Wqo-Theory) and several fields*of*Computer Science (Verification and Termination*of*Infinite-State Systems, Automata and Formal Languages, Term Rewriting and Proof Theory ...*One**of*the open challenges is the strength*of**Fraïssé's**order*type*conjecture*. ...##
###
A survey of homogeneous structures

2011
*
Discrete Mathematics
*

The structure (Q, <), where < is the usual

doi:10.1016/j.disc.2011.01.024
fatcat:fqpfnc25hjdajekkiyaeszw6ye
*order**on*the rationals, is homogeneous. ... However the standard method*of*construction*of*homogeneous structures, described next, is by*Fraissé's*Theorem. ... They*conjecture*that this holds*for*all*finitely*bounded ω-categorical homogeneous structures. ...##
###
Embeddability on functions: order and chaos
[article]

2018
*
arXiv
*
pre-print

We study the quasi-

arXiv:1802.08341v1
fatcat:sihxpvi6zjbbdmwv7fuvyzhwve
*order**of*topological embeddability*on*definable functions between Polish zero-dimensional spaces. ... We prove that no Baire class admits a maximal element, except*for*the class*of*continuous functions which admits a maximum element. ... We would like to thank the Descriptive Set Theory group at Paris 6 and the Turin logic group*for*their patience and careful attention during the seminar presentation*of*the material exposed in this paper ...##
###
On better-quasi-ordering classes of partial orders
[article]

2014
*
arXiv
*
pre-print

We provide a method

arXiv:1408.0315v3
fatcat:xhdo5cv25jg6hl5jq7v45c32xu
*of*constructing better-quasi-*orders*by generalising a technique*for*constructing operator algebras that was developed by Pouzet. ... This generalises theorems*of*Laver, Corominas and Thomassé regarding σ-scattered*linear**orders*and trees, countable forests and N-free partial*orders*respectively. ... Acknowledgements The author thanks Mirna Džamonja, Yann Pequignot and Raphaël Carroy*for*extremely helpful discussions and remarks. ...##
###
Multicoloured Random Graphs: Constructions and Symmetry
[article]

2014
*
arXiv
*
pre-print

The large number

arXiv:1406.7870v2
fatcat:othjxv7jzbhq7pupvjfmonunva
*of*references should help in making this a resource*for*anyone interested in beginning research in this or allied fields. ... This is a research monograph*on*constructions*of*and group actions*on*countable homogeneous graphs, concentrating particularly*on*the simple random graph and its edge-coloured variants. ...*For*any g ∈ Aut(L), at most O(n d−1 ) points are fixed by g, since xg = x is a set*of**linear*equations with non-zero*rank*. ...##
###
Complexity Classification in Infinite-Domain Constraint Satisfaction
[article]

2019
*
arXiv
*
pre-print

We demonstrate the feasibility

arXiv:1201.0856v10
fatcat:t32gozkmqrauze2gzqn2ze5kaa
*of*our approach with two complete complexity classification results:*one**on*CSPs in temporal reasoning, the other*on*a generalization*of*Schaefer's theorem*for*propositional ... A constraint satisfaction problem (CSP) is a computational problem where the input consists*of*a*finite*set*of*variables and a*finite*set*of*constraints, and where the task is to decide whether there exists ... I want to thank my institution, the CNRS,*for*the great ...##
###
Aspects of infinite permutation groups
[chapter]

*
Groups St Andrews 2005
*

I shall concentrate

doi:10.1017/cbo9780511721212.002
fatcat:h4ecigjgvbc57dfewovhazxqom
*on*a few topics, following the pattern*of*my conference lectures: the random graph (a case study); homogeneous relational structures (a powerful construction technique*for*interesting ... There were a few papers,*for*example [10, 62] , and a set*of*lecture notes by Wielandt [72], from the 1950s. ... Theorem 3.5*For*any k ≥ 1, the group*of**order*-automorphisms*of*the rational numbers contains a free subgroup F*of*countable*rank*with the properties that*for*any two k-tuples x 1 < x 2 < · · · < x k and ...
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