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The result provides a far-reaching generalization of an important classical result of Kleitman.\n  The well-known Erd\\H os Matching Conjecture suggests the largest size of a family $\\mathcal F\\subset {[n]\\choose k}$ with no $s$ pairwise disjoint sets. 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The result provides a far-reaching generalization of an important classical result of Kleitman.\n  The well-known Erd\\H os Matching Conjecture suggests the largest size of a family $\\mathcal F\\subset {[n]\\choose k}$ with no $s$ pairwise disjoint sets. 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