Partitioning Subgraphs of Profinite Ordered Graphs

Abstract

Let K be the class of all inverse limits \(G = {\underleftarrow {\lim }_{n \in \mathbb{N}}}{G_n}\), where each Gn is a finite ordered graph. GK is universal if every BK embeds continuously into G.

Theorem (1). For every finite ordered graph A there exists a least natural number k(A)≥1 such that for every universal GK, for every finite Baire measurable partition of the set \(\left( \begin{gathered} G \hfill \\ A \hfill \\ \end{gathered} \right)\) of all copies of A in G, there is a closed copy G′⊆G of G such that \(\left( \begin{gathered} G' \hfill \\ A \hfill \\ \end{gathered} \right)\) meets at most k(A) parts. In the arrow notation:

$$G \to Baire\left( G \right)_{ < \infty |k\left( A \right)}^A$$

.

Theorem (2). The probability that k(A)=1, for a finite ordered graph A, chosen randomly with uniform probability from all graphs on {0,1,...,n–1}, tends to 1 as n grows to infinity, where k(A) is the number given by Theorem (1).

As a corollary

Theorem (3). The class K with Baire partitions satisfies with high probability the A-partition property for a finite ordered graph A, where the A-partition property is

$$\left( {{\forall _{B \in K}}} \right)\left( {{\exists _{C \in K}}} \right)C \to _{Baire}{\left( B \right)^A}$$

.

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Correspondence to Stefan Geschke.

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Partially supported by a German-Israel Foundation for Scientific Research & development grant. The second author was also supported by the DAAD project 57156702, “Universal profinite graphs”. The last author was also supported by an Israel Science Foundation grant number 1365/14.

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Huber, S., Geschke, S. & Kojman, M. Partitioning Subgraphs of Profinite Ordered Graphs. Combinatorica 39, 659–678 (2019). https://doi.org/10.1007/s00493-018-3479-9

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Mathematics Subject Classification (2010)

  • Primary 05D10, 05C15
  • Secondary: 05C55, 03C15, 04E45