Partitioning subgraphs of profinite ordered graphs (Q2322548)

Let \(\mathcal{K}\) be the class of all inverse limits \(G=\lim_{\leftarrow \, n\in \mathbb{N}}G_{n}\), where each \(G_n\) is a finite ordered graph. \(G\in \mathcal{K}\) is said to be universal if every \(B\in \mathcal{K}\) embeds continously into \(G\). In this paper, it is shown that for every finite ordered graph \(A\) there exists a least natural number \(k(A)\geq 1\) such that for every universal \(G\in \mathcal{K}\), for every finite Baire measurable partition of the set \(\binom{G}{A}\) of all copies of \(A\) in \(G\), there is a closed copy \(G^\prime\subseteq G\) of \(G\) such that \(\binom{G^\prime}{A}\) meets at most \(k(A)\) parts. Moreover, the probability that \(k(A)=1\), for a finite ordered graph \(A\), chosen randomly with uniform probability from all graphs on \(\{ 0,\ldots ,n-1\}\), tends to 1 as \(n\) grows to infinity and the class \(\mathcal{K}\) with Baire partitions satisfies with high probability the following partition property for a finite ordered graph \(A\): for every inverse limit of finite ordered graphs \(B\) there exists an inverse limit of finite ordered graph \(C\) such that for every Baire measurable partition of \(\binom{C}{A}\) to finitely many parts there is a closed copy of \(B\) in \(C\) such that \(\binom{B}{A}\) meets exactly one of the parts.