The 42 reducts of the random ordered graph

Bodirsky, Manuel; Pinsker, Michael; Pongrácz, András

doi:10.1112/plms/pdv037

Cited by 19 publications

(19 citation statements)

References 16 publications

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“…We remark that in our case, the structure ∆ above is the random ordered graph (roughly the random graph equipped with the order of the rationals in a random way -confer Section 7) rather than the random graph G itself. The reducts of this structure have recently been classified [13].…”

Section: Discussion Of Our Strategymentioning

confidence: 99%

Schaefer's Theorem for Graphs

Bodirsky

Pinsker

2015

J. ACM

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Abstract. Schaefer's theorem is a complexity classification result for so-called Boolean constraint satisfaction problems: it states that every Boolean constraint satisfaction problem is either contained in one out of six classes and can be solved in polynomial time, or is NP-complete.We present an analog of this dichotomy result for the propositional logic of graphs instead of Boolean logic. In this generalization of Schaefer's result, the input consists of a set W of variables and a conjunction Φ of statements ("constraints") about these variables in the language of graphs, where each statement is taken from a fixed finite set Ψ of allowed quantifier-free first-order formulas; the question is whether Φ is satisfiable in a graph.We prove that either Ψ is contained in one out of 17 classes of graph formulas and the corresponding problem can be solved in polynomial time, or the problem is NP-complete. This is achieved by a universal-algebraic approach, which in turn allows us to use structural Ramsey theory. To apply the universal-algebraic approach, we formulate the computational problems under consideration as constraint satisfaction problems (CSPs) whose templates are first-order definable in the countably infinite random graph. Our method for classifying the computational complexity of those CSPs is based on a Ramsey-theoretic analysis of functions acting on the random graph, and we develop general tools suitable for such an analysis which are of independent mathematical interest.

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Section: Discussion Of Our Strategymentioning

confidence: 99%

Schaefer's Theorem for Graphs

Bodirsky

Pinsker

2015

J. ACM

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“…Quite surprisingly, countable structures Γ that are homogeneous in a finite relational language tend to have finitely many reducts, up to interdefinability (see e.g. [3,10,13,42,43,52,53]), and Thomas [52] conjectured that this is always the case. If the age of Γ is Ramsey, or a homogeneous expansion of the structure is Ramsey, then this helps in classifying reducts; we refer to the survey article [9] for the technical details.…”

Section: The Ordering Propertymentioning

confidence: 99%

Ramsey classes: examples and constructions

Bodirsky¹

2015

Surveys in Combinatorics 2015

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This article is concerned with classes of relational structures that are closed under taking substructures and isomorphism, that have the joint embedding property, and that furthermore have the Ramsey property, a strong combinatorial property which resembles the statement of Ramsey's classic theorem. Such classes of structures have been called Ramsey classes. Nešetřil and Rödl showed that they have the amalgamation property, and therefore each such class has a homogeneous Fraïssé limit. Ramsey classes have recently attracted attention due to a surprising link with the notion of extreme amenability from topological dynamics. Other applications of Ramsey classes include reduct classification of homogeneous structures.We give a survey of the various fundamental Ramsey classes and their (often tricky) combinatorial proofs, and about various methods to derive new Ramsey classes from known Ramsey classes. Finally, we state open problems related to a potential classification of Ramsey classes.

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“…The proof of this highly non-trivial result applies all existing tricks and techniques. Similarly to [5], where the 42 proper reducts of the ordered random graph is determined on 42 pages. In [5] it is mentioned that we do not even know how to show that the lattice of reducts has only finitely many atoms, or no infinite ascending chains.…”

Section: Introductionmentioning

confidence: 99%

“…Similarly to [5], where the 42 proper reducts of the ordered random graph is determined on 42 pages. In [5] it is mentioned that we do not even know how to show that the lattice of reducts has only finitely many atoms, or no infinite ascending chains. In order to learn more, it seems to be unavoidable to verify the conjecture for more of the classical structures from model theory, independently from whether or not we believe the conjecture.…”

Section: Introductionmentioning

confidence: 99%