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We shall prove the following partition theorems: (1) For every set S and for each cardinal κ{script} ≥ ω, |S| ≥ κ{script} there exists a partition T: [S]^κ{script} → 2^κ{script} such that for every pairwise disjoint familie[Figure not available: see fulltext.] and every α < 2^κ{script} there exists a set[Figure not available: see fulltext.] (2) Suppose κ{script} ≥ ω, 2^<κ{script} and S an arbitrary set, 2^|S| ≤ (2^κ{script})^+ω Then there exists a partition T: P(S) → 2^κ{script} such that for every pairwise disjoint family[Figure not available: see fulltext.] and every α < 2^κ{script} there exists a set[Figure not available: see fulltext.] Both theorems will give partial answers to an Erdo{combining double acute accent}s problem.