Some Erdos-Ko-Rado theorems for injections

This paper investigates t-intersecting families of injections. where two inject ions a, b from vertical bar k vertical bar to vertical bar n vertical bar t-intersect if there exists X subset of vertical bar k vertical bar with vertical bar X vertical bar >= t such that a(X) = a(X) for all x is an element of X. We prove that if F is a 1-intersecting injection family of maximal size then all elements of F have a fixed image point in common. We show that when n is large in terms of k and t, the set of injections which fix the first t points is the only t-intersecting injection family of maximal size, up to permutaions of vertical bar k vertical bar and vertical bar n vertical bar. This is not the case for small it. Indeed. we prove that if k is large in terms of k - t and it - k, the largest t-intersecting injection families are obtained from a process of Saturation rather than fixing. (C) 2009 Elsevier Ltd. All rights reserved.