Simultaneous sequential compactness

Given a cardinal lambda, we say that a topological space is lambda-simultaneously sequentially compact (lambda-ssc) if for each family of lambda-many sequences in the space there exists an infinite subset H of the index set omega such that each sequence in the family converges when restricted to H. Our main question is: given a sequentially compact space, which are the cardinals lambda such that our space is lambda-ssc? Our answers will depend on two cardinals, known as splitting number and distributivity number, as well as the weight of the space. Under the further assumption that the space is Hausdorff, we introduce a new cardinal invariant, which takes the place of the weight to improve our main result. This is joint work with Cesare Straffelini.