Abstract
(1) VERTICAL LINES. The answer depends, of course, on the meaning one attaches to many and large. The natural meaning of many involves cardinal numbers so that, for instance, a collection could be said to have many members if it is uncountable, or if it is just infinite, or even if it is just not empty. If large is interpreted to have the same (cardinal-number) meaning, then the answer to the question is no. Example: the vertical lines in the plane constitute an uncountable collection of uncountable sets such that the intersection of every subcollection with more than one element is as small as possible, namely, empty. Since set-theoretically the plane and the line are the same, it is easy to produce a similar example in the line: there exists an uncountable collection of pairwise disjoint uncountable subsets of, say, the unit interval.
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