On the number of latent subsets of intersecting collections

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

$\text{?}{}_1{}^{\mathrm{L}}\text{?}{}_2{}^{\mathrm{L}}\text{? ?}{}_1\text{?}{}_2$

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.