The complete characterization of the minimum size supertail in a subspace partition

Let q be a prime power and let n be a positive integer. Let V = V(n, q) denote the vector space of dimension n over F-q. A subspace partition P of V is a collection of subspaces of V with the property that each nonzero vector is in exactly one of the subspaces in P. Suppose that d(1),..., d(k) are the different dimensions, in increasing order, that occur in the subspace partition P. For any integer s, with 2 <= s <= k, the d(s)-supertail S of P is the collection of all subspaces X is an element of P such that dim X < d(s). It was shown that vertical bar S vertical bar >=sigma(q)(ds,ds--1), where sigma(q)(d(s), d(s-1)) denotes the minimum number of subspaces over all subspace partitions of V (d(s), q) in which the largest subspace has dimension d(s-1). Moreover, it was shown that if d(s) >= 2d(s-1) and equality holds in the previous bound on vertical bar S vertical bar, then the union of the subspaces in S forms a subspace. This characterization was also conjectured to hold if d(s) < 2d(s-1). This conjecture was recently proved in certain cases. In this paper, we use a much simpler approach to completely settle this conjecture. (C) 2018 Elsevier Inc. All rights reserved.