It is proved that every normalized weakly null sequence has a subsequence which is convexly unconditional. Further, an Hierarchy of summability methods is introduced and with this we give a complete classification of the complexity of weakly null sequences.

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Abstract.Let A' be a compact Hausdorff space and k its Souslin number.2 We prove that if a is a cardinal such that either a and cf(a) are greater than k and strongly «-inaccessible or else a is regular and greater than k, then X has (a,%a) caliber.Restricting our interest to the category of compact spaces X with S{ X) = u+ (i.e. X satisfy the countable chain condition), the above statement takes, under G.C.H., the following form.For any compact space X with S( X) = u+ , we have that (a) if a is a cardinal and cf(a) does not have the form ß + with cf(/3) = u, then a is caliber for the space X.(b) If e = ß+ and cf(j8) = a then (a, ß) is caliber for X.A related example shows that the result of (b) is in a sense the best possible.Introduction. Let X be a topological space and a, ß cardinals. We say that the pair (a, ß) is caliber for X (or X has (a, ß) caliber) if every family {Ff: £ < a) of nonempty open subsets of X has a subfamily {V(: £ E B) such that D ies Ff ^ 0

Abstract. An elementary setting of the classical Ramsey property is given, which leads to simple proofs of the relevant theorems of Galvin-Prikry and Silver.

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