Finiteness spaces, graphs and "coherence"

We look at a sub-collection of finiteness spaces introduced in [2] based on the notion of coherence spaces from [4]. The original idea was to generalize the notion of stable functions between coherence spaces to interpret the algebraic lambda-calculus ([6]) or even the differential lambda-calculus ([3]). An important tool for this analysis is the infinite Ramsey theorem. 0. Introduction. The category of coherence spaces was the first denotational model for linear logic (see [4]): the basic objects are reflexive, non oriented graphs; and we are more specifically interested by their cliques (complete subgraph). If C is such a graph, we write C(C) for the collection of its cliques. Coherence spaces enjoy a very rich algebraic structure where the most important operations on them are: • taking the (reflexive closure of the) complement (written C ⊥ 1); • taking a cartesian product (written C 1 ⊗ C 2); • taking a disjoint union (written C 1 ⊕ C 2). If one only looks at the vertices, the corresponding operations are simply the identity, the usual cartesian product " × " and the disjoint union " ⊕ ". More recently, Thomas Ehrhard introduced the notion of finiteness spaces ([2]) to give a model to the differential λ-calculus ([3]), which can be seen as an enrichment of linear logic. The point that interests us most here is that thue collection of finitary sets of a finiteness space are closed under finite sums (i.e. finite unions) to take into account a notion of " non-deterministic sum " of terms. (See also [6].) This is definitely not true of cliques of a coherence space...