383 research outputs found
Grid classes and the Fibonacci dichotomy for restricted permutations
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
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
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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