On the number of latent subsets of intersecting collections

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Abstract

Given two collections F1 and F2 of sets, each member of one intersecting each member of the other, let the collections of latent sets FiL, i = 1, 2, be the sets that are contained in members of Fi but that are not themselves members of Fi. If lower case letters indicate the size of the collections we then have ƒ1Lƒ2L ⩾ ƒ1ƒ2

This result is used to prove that a self-intersecting subfamily F of a simplical complex G having the property that any element of F contains s1 or s2 can be no larger than the lesser of the number of elements of G containing s1 and the number containing s2. Certain extensions and a related conjecture of Chvátal are described.

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Supported in part by O.N.R. Contract N00014-67-A-0204-0063.

Supported in part by the U.S. Army Office (Durham) under contract DAHCO4-70-C-0058.