383 research outputs found

    Grid classes and the Fibonacci dichotomy for restricted permutations

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    We introduce and characterise grid classes, which are natural generalisations of other well-studied permutation classes. This characterisation allows us to give a new, short proof of the Fibonacci dichotomy: the number of permutations of length n in a permutation class is either at least as large as the nth Fibonacci number or is eventually polynomial

    The infinite random simplicial complex

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    We study the Fraisse limit of the class of all finite simplicial complexes. Whilst the natural model-theoretic setting for this class uses an infinite language, a range of results associated with Fraisse limits of structures for finite languages carry across to this important example. We introduce the notion of a local class, with the class of finite simplicial complexes as an archetypal example, and in this general context prove the existence of a 0-1 law and other basic model-theoretic results. Constraining to the case where all relations are symmetric, we show that every direct limit of finite groups, and every metrizable profinite group, appears as a subgroup of the automorphism group of the Fraisse limit. Finally, for the specific case of simplicial complexes, we show that the geometric realisation is topologically surprisingly simple: despite the combinatorial complexity of the Fraisse limit, its geometric realisation is homeomorphic to the infinite simplex.Comment: 33 page

    Kernelizing MSO Properties of Trees of Fixed Height, and Some Consequences

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    Fix an integer h>=1. In the universe of coloured trees of height at most h, we prove that for any graph decision problem defined by an MSO formula with r quantifiers, there exists a set of kernels, each of size bounded by an elementary function of r and the number of colours. This yields two noteworthy consequences. Consider any graph class G having a one-dimensional MSO interpretation in the universe of coloured trees of height h (equivalently, G is a class of shrub-depth h). First, class G admits an MSO model checking algorithm whose runtime has an elementary dependence on the formula size. Second, on G the expressive powers of FO and MSO coincide (which extends a 2012 result of Elberfeld, Grohe, and Tantau)
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