Abstract topological Ramsey theory for nets

By introducing the class of coideals on an infinite directed set X, a class that contains all the adequate ultrafilters on X, it becomes possible to prove partition results for the ordered finite or infinite sequences in X with respect to a given coideal on X. Our theory extends the classical topological Ramsey theory, and in addition includes as particular cases (a) the corresponding theory for coideals on the set of natural numbers proved by Louveau, Mathias, Farah and Todorcevic, (b) the Milliken–Taylor partition theorems for sequences of finite subsets of natural numbers, and (c) the partition theorems for sequences of words proved by Carlson, Bergelson–Blass–Hindman, Farmaki.