Combinatorial Optimization Problems for Systems of Subsets

This paper discusses conjectures based on the combinatorial optimization problem of maximizing w(F) over a specified collection of families F of subsets of {1,...,n}, where w is defined first on the subsets A,B,... of {1,...,n} and then extended additively by w (F) = Sigma A EpsilonF w(A). Most of the conjectures involve the collection M of maximal intersecting systems and its subcollection I of families generated by singleton subsets of {1,...,n}.