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Abstract: The Kechris-Pestov-Todorcevic (KPT) correspondence provides a deep connection between the structural Ramsey properties of a class $\mathcal{K}$ and the dynamical properties (extreme amenability) of its automorphism group. However, a full generalization of this theory for arbitrary Polish groups, or even for all linear Polish groups (isometry groups of Banach spaces), remains open. Key concepts like ``big Ramsey degrees'' (the structure of types) or ``small Ramsey degrees'' (factorization theorems) are not well understood in this general metric setting. 1. \textbf{(Analogues of Big Ramsey)} We will first discuss $\omega$-categorical Banach spaces. We characterize the orbit spaces (spaces of complete types) for spaces like $C(K, X)$ and $L_p([0,1], X)$. This study of the compactness and structure of the space of types may serve as a linear analogue to the study of ``big Ramsey'' properties. \par 2. \textbf{(Analogues of Small Ramsey)} Second, we will discuss metric versions of the Dual Ramsey Theorem. [cite_start]We present factorization theorems for compact colorings of matrices and Grassmannians over $\mathbb{R}$ and $\mathbb{C}$. This work, which was partially developed during the BIRS workshop "Homogeneous Structures" (Nov. 8-13, 2015), somehow provides an analogue of ``small Ramsey degrees'' for these linear structures. \bigskip \noindent\textit{The first part is joint work with V. Ferenczi and V. Olmos-Prieto. The second part is joint work with D. Barto\v{s}ov\'a, M. Lupini, and B. Mbombo.}