This short note presents a new relation between coherent spaces and finiteness spaces. This takes the form of a functor from COH to FIN commuting with the additive and multiplicative structure of linear logic. What makes this correspondence possible and conceptually interesting is the use of the infinite Ramsey theorem. Along the way, the question of the cardinality of the collection of finiteness spaces on N is answered. Basic knowledge about coherent spaces and finiteness spaces is assumed.

• 03F52 - Proof-theoretic aspects of linear logic and other substructural logics
• 03G30 - Categorical logic, topoi