We study how homogeneous ideals in the exterior algebra ? V over a finite-dimensional vector space V are minimally generated. In particular, we solve the following problems: Starting with an element pv of degree v, what is the maximum length l of a sequence pv, . . . , pv+l-l, with degpi = i, and such that pi is not in the ideal generated by pl, . . . , pi-l? What is the maximal possible number of minimal generators of degree d of a homogeneous ideal which does not contain all elements of degree d + 1? Our main tool is the Kruskal-Katona theorem.