## Abstract

Let *K* be the class of all inverse limits \(G = {\underleftarrow {\lim }_{n \in \mathbb{N}}}{G_n}\), where each *G*_{n} 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 \(\left( \begin{gathered} G \hfill \\ A \hfill \\ \end{gathered} \right)\) 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:

.

**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

.

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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