We prove an infinite-dimensional version of an approximate Ramsey theorem of Gowers, initially used to show that every Lipschitz function on the unit sphere of c 0 is oscillation stable. To do so, we use the theory of ultra-Ramsey spaces developed by Todorcevic in order to obtain an Ellentuck-type theorem for the space of all infinite block sequences in FIN ±k .
Despite significant progress in the study of big Ramsey degrees, the big Ramsey degrees of many classes of structures with finite small Ramsey degrees remain unknown. In this paper, we investigate the big Ramsey degrees of unrestricted relational structures in (possibly) infinite languages and demonstrate that they have finite big Ramsey degrees if and only if there are only finitely many relations of every arity. This is the first time that the finiteness of big Ramsey degrees has been established for an infinite-language random structure.
We show that the infinite-dimensional versions of Gowers' FIN k and FIN ±k theorems can be parametrized by an infinite sequence of perfect subsets of 2 ω . To do so, we use ultra-Ramsey theory to obtain exact and approximate versions of a result which combines elements from both Gowers' theorems and the Hales-Jewett theorem. As a consequence, we obtain a parametrized version of Gowers' c 0 theorem.
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