Homogeneous Dual Ramsey Theorem

For positive integers $k < n$ such that $k$ divides $n$, let $(n)^k_{\hom}$ be the set of homogeneous $k$-partitions of $\{1, \dots, n\}$, that is, the set of partitions of $\{1, \dots, n\}$ into $k$ classes of the same cardinality. In the article "Ramsey properties of infinite measure algebras and topological dynamics of the group of measure preserving automorphisms: some results and an open problem" by Kechris, Sokic, and Todorcevic, the following question was asked: Is it true that given positive integers $k < m$ and $N$ such that $k$ divides $m$, there exists a number $n>m$ such that $m$ divides $n$, satisfying that for every coloring $(n)^k_{\hom}=C_1\cup\dots\cup C_N$ we can choose $u\in (n)^m_{\hom}$ such that $\{t\in (n)^k_{\hom}: t\mbox{ is coarser than } u\}\subseteq C_i$ for some $i$? In this note we give a positive answer to that question. This result turns out to be a homogeneous version of the finite Dual Ramsey Theorem of Graham-Rothschild. As explained by Kechris, Sokic, and Todorcevic in their article, our result also proves that the class $\mathcal{OMBA}_{\mathbb Q_2}$ of naturally ordered finite measure algebras with measure taking values in the dyadic rationals has the Ramsey property.