TY - JOUR

T1 - Partitioning Subgraphs of Profinite Ordered Graphs

AU - Huber, Stefanie

AU - Geschke, Stefan

AU - Kojman, Menachem

N1 - Publisher Copyright:
© 2018, János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg.

PY - 2019/6/1

Y1 - 2019/6/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85051735761&partnerID=8YFLogxK

U2 - 10.1007/s00493-018-3479-9

DO - 10.1007/s00493-018-3479-9

M3 - Article

AN - SCOPUS:85051735761

SN - 0209-9683

VL - 39

SP - 659

EP - 678

JO - Combinatorica

JF - Combinatorica

IS - 3

ER -