Some Erdős–Ko–Rado theorems for injections

This paper investigates tt-intersecting families of injections, where two injections a,ba,b from [k][k] to [n]t-intersect if there exists X⊆[k]X⊆[k] with |X|≥t|X|≥t such that a(x)=b(x)a(x)=b(x) for all x∈Xx∈X. We prove that if FF is a 1-intersecting injection family of maximal size then all elements of FF have a fixed image point in common. We show that when nn is large in terms of kk and tt, the set of injections which fix the first tt points is the only tt-intersecting injection family of maximal size, up to permutations of [k][k] and [n][n]. This is not the case for small nn. Indeed, we prove that if kk is large in terms of k−tk−t and n−kn−k, the largest tt-intersecting injection families are obtained from a process of saturation rather than fixing.