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Statistics of orderings

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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 \(\overrightarrow{a}\) 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 \(\overrightarrow{a}\). This for a particular choice of \(\overrightarrow{a}\) 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.

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Acknowledgements

Many thanks to the referee for detailed reading and numerous helpful comments.

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Authors and Affiliations

  1. IUUK MFF, Charles University, Prague, Czech Republic

    Jaroslav Nešetřil

  2. Emory University, Atlanta, GA, USA

    Vojtěch Rödl

Authors
  1. Jaroslav Nešetřil
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  2. Vojtěch Rödl
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Correspondence to Jaroslav Nešetřil.

Additional information

J. Nešetřil Partially supported by the Project LL1201 ERC-CZ CORES and by CE-ITI P202/12/G061 of GAČR. V. Rödl Partially supported by the project LL1201 ERC-CZ CORES and by CE-ITI P202/12/G061 of GAČR and by NSF Grands DMS 1102086 and DMS 1301698.

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Nešetřil, J., Rödl, V. Statistics of orderings. Abh. Math. Semin. Univ. Hambg. 87, 421–433 (2017). https://doi.org/10.1007/s12188-016-0174-x

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  • DOI: https://doi.org/10.1007/s12188-016-0174-x

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