  Let r >= s >= 0 be integers and G be an r-graph. The higher inclusion matrix
M_s^r(G) is a {0,1}-matrix with rows indexed by the edges of G and columns
indexed by the subsets of V(G) of size s: the entry corresponding to an edge e
and a subset S is 1 if S is contained in e and 0 otherwise. Following a
question of Frankl and Tokushige and a result of Keevash, we define the
rank-extremal function rex(n,t,r,s) as the maximum number of edges of an
r-graph G having rank M_s^r(G) <=\binom{n}{s} - t. For t at most linear in n we
determine this function as well as the extremal r-graphs. The special case t=1
answers a question of Keevash.
