Concerning the shape of a geometric lattice
  A well-known conjecture states that the Whitney numbers of the second kind of a geometric lattice (simple matroid) are logarithmically concave. We show this conjecture to be equivalent to proving an upper bound on the number of new copoints in the free erection of the associated simple matroid M. A bound on the number of these new copoints is given in terms of the copoints and colines of M. Also, the points-lines-planes conjecture is shown to be equivalent to a problem concerning the number subgraphs of a certain bipartite graph whose vertices are the points and lines of a geometric lattice. 
W. M. B. Dukes