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Answering a 25-year-old problem of Erdos and Gallai, we prove that if a set system Z of rank r cannot be represented by I elements, then there is a subsystem HO' s Z on less than ( :I! I) + (':I; ') elements that cannot be represented by I elements either. Apart from a constant factor, this upper bound is best possible for every r and f. Moreover, we determine the maximum cardinality of strongly independent vertex sets in r-critical and intersecting v-critical hypergraphs of given rank, and describe the extremal structures.doi:10.1016/0097-3165(89)90064-2 fatcat:wrhrevzwejadbf5z2qv6tlmz7e