  Let "ex" be the cardinality of the smallest independent family of subsets of
omega (independent means that all nontrivial Boolean combinations are infinite)
which cannot be extended to a homogeneous independent family. "Homogeneous"
means that every finite partial map (points to points, sets to sets) that is
compatible with the family is induced by a permutation of omega.
  Despite its apparently ``complex'' (second order) definition, this cardinal
is equal to a well-known ``simple'' cardinal invariant from Cichon's diagram.
