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

Alberto Marcone, Antonio Montalbán
2009 Annals of Pure and Applied Logic  
We then show that 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.  ... 
doi:10.1016/j.apal.2009.01.007 fatcat:mf7iar5vbrawfkxybpxi547744

Up to equimorphism, hyperarithmetic is recursive

Antonio Montalbán
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  ... 
doi:10.2178/jsl/1120224717 fatcat:2ef664vb5ngxfifyputfezj6ja

The reverse mathematics of wqos and bqos [article]

Alberto Marcone
2019 arXiv   pre-print
The classification from the reverse mathematics viewpoint 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.  ... 
arXiv:1707.08365v5 fatcat:ym22ayu7ibhdtfrqnf2pi5ko44

Laver and set theory

Akihiro Kanamori
2016 Archive for Mathematical Logic  
In this commemorative article, the work 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  ... 
doi:10.1007/s00153-015-0462-7 fatcat:5ziwvh27arbprhsdlmku4f2npe

Countably Complementable Linear Orderings

Antonio Montalbán
2007 Order  
Using similar methods and introducing the notion 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.  ... 
doi:10.1007/s11083-006-9049-6 fatcat:eur6hi7wjbg2ljebi3uamryyhe

Open Questions in Reverse Mathematics

Antonio Montalbán
2011 Bulletin of Symbolic Logic  
We present a list 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.  ... 
doi:10.2178/bsl/1309952320 fatcat:maim2j6vovfcpjjkrzm7xeicqi

Equimorphy -- The Case of Chains [article]

C. Laflamme, M. Pouzet, R.Woodrow
2014 arXiv   pre-print
Such structures cannot be expected to be isomorphic, and in this paper we investigate the special case 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.  ... 
arXiv:1407.2894v1 fatcat:r7x5ijysgvf7tchkikgqa4s7ve

The maximal linear extension theorem in second order arithmetic

Alberto Marcone, Richard A. Shore
2011 Archive for Mathematical Logic  
We show that the maximal 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  ... 
doi:10.1007/s00153-011-0231-1 fatcat:nau46smsnzczpojbwq5ik2vqqy

Well Quasi-Orders in Computer Science (Dagstuhl Seminar 16031)

Jean Goubault-Larrecq, Monika Seisenberger, Victor Selivanov, Andreas Weiermann, Marc Herbstritt
2016 Dagstuhl Reports  
, topological complexity 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.  ... 
doi:10.4230/dagrep.6.1.69 dblp:journals/dagstuhl-reports/Goubault-Larrecq16 fatcat:o3uwzu5ptfavfes6kgexpi4a6q

A survey of homogeneous structures

Dugald Macpherson
2011 Discrete Mathematics  
The structure (Q, <), where < is the usual 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.  ... 
doi:10.1016/j.disc.2011.01.024 fatcat:fqpfnc25hjdajekkiyaeszw6ye

Embeddability on functions: order and chaos [article]

Raphaël Carroy, Yann Pequignot, Zoltán Vidnyánszky
2018 arXiv   pre-print
We study the quasi-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  ... 
arXiv:1802.08341v1 fatcat:sihxpvi6zjbbdmwv7fuvyzhwve

On better-quasi-ordering classes of partial orders [article]

Gregory McKay
2014 arXiv   pre-print
We provide a method 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.  ... 
arXiv:1408.0315v3 fatcat:xhdo5cv25jg6hl5jq7v45c32xu

Multicoloured Random Graphs: Constructions and Symmetry [article]

Sam Tarzi
2014 arXiv   pre-print
The large number 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.  ... 
arXiv:1406.7870v2 fatcat:othjxv7jzbhq7pupvjfmonunva

Complexity Classification in Infinite-Domain Constraint Satisfaction [article]

Manuel Bodirsky
2019 arXiv   pre-print
We demonstrate the feasibility 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  ... 
arXiv:1201.0856v10 fatcat:t32gozkmqrauze2gzqn2ze5kaa

Aspects of infinite permutation groups [chapter]

Peter J. Cameron, C. M. Campbell, M. R. Quick, E. F. Robertson, G. C. Smith
Groups St Andrews 2005  
I shall concentrate 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  ... 
doi:10.1017/cbo9780511721212.002 fatcat:h4ecigjgvbc57dfewovhazxqom
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