Some Erdős–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.