Partitioning Subgraphs of Profinite Ordered Graphs

Stefanie Huber, Stefan Geschke, Menachem Kojman

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

Let K be the class of all inverse limits G=lim←n∈ℕGn, where each Gn is a finite ordered graph. G∈K is universal if every B∈K 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 G∈K, for every finite Baire measurable partition of the set (GA) of all copies of A in G, there is a closed copy G′⊆G of G such that (G'A) meets at most k(A) parts. In the arrow notation: G→Baire(G)<∞|k(A)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 (∀B∈K)(∃C∈K)C→Baire(B)A.

Original languageEnglish
Pages (from-to)659-678
Number of pages20
JournalCombinatorica
Volume39
Issue number3
DOIs
StatePublished - 1 Jun 2019

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Computational Mathematics

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