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Statistics of orderings

Statistics of orderings In this paper, we study a Ramsey type problems dealing with the number of ordered subgraphs present in an arbitrary ordering of a larger graph. Our first result implies that for every vertex ordered graph G on k vertices and any stochastic vector $$\overrightarrow{a}$$ a → with k! entries, there exists a graph H with the following property: for any linear order of the vertices of H, the number of induced ordered copies of G in H is asymptotically equal to a convex combination of the entries in $$\overrightarrow{a}$$ a → . This for a particular choice of $$\overrightarrow{a}$$ a → yeilds an earlier result of Angel, Lyons, and Kechris. We also consider a similar question when the ordering of vertices is replaced by the ordering of pairs of vertices. This problem is more complex problem and we prove some partial results in this case. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Springer Journals

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References (11)

Publisher
Springer Journals
Copyright
Copyright © 2016 by The Author(s)
Subject
Mathematics; Mathematics, general; Algebra; Differential Geometry; Number Theory; Topology; Geometry
ISSN
0025-5858
eISSN
1865-8784
DOI
10.1007/s12188-016-0174-x
Publisher site
See Article on Publisher Site

Abstract

In this paper, we study a Ramsey type problems dealing with the number of ordered subgraphs present in an arbitrary ordering of a larger graph. Our first result implies that for every vertex ordered graph G on k vertices and any stochastic vector $$\overrightarrow{a}$$ a → with k! entries, there exists a graph H with the following property: for any linear order of the vertices of H, the number of induced ordered copies of G in H is asymptotically equal to a convex combination of the entries in $$\overrightarrow{a}$$ a → . This for a particular choice of $$\overrightarrow{a}$$ a → yeilds an earlier result of Angel, Lyons, and Kechris. We also consider a similar question when the ordering of vertices is replaced by the ordering of pairs of vertices. This problem is more complex problem and we prove some partial results in this case.

Journal

Abhandlungen aus dem Mathematischen Seminar der Universität HamburgSpringer Journals

Published: Jan 4, 2017

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