Some Erdos-Ko-Rado theorems for injections

This paper investigates t-intersecting families of injections, where two injections a, b from [k] to [n] t-intersect if there exists X ⊆ [k] with |X| ≥ t such that a(x) = b(x) for all x ∈ 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 permutations of [k] and [n]. This is not the case for small n. Indeed, we prove that if k is large in terms of k − t and n− k, the largest t-intersecting injection families are obtained from a process of saturation rather than fixing.