Colorful flowers

For a set A let k[A] denote the family of all k-element subsets of A. A function f:k[A]→C is a local coloring if it maps disjoint sets of A into different elements of C. A family F⊆k[A] is called a flower if there exists E∈[A]k−1 so that |F∩F′|=E for all F,F′∈F, F≠F′. A flower is said to be colorful if f(F)≠f(F′) for any two F,F′∈F. In the paper we find the smallest cardinal γ such that there exists a local coloring of k[A] containing no colorful flower of size γ. As a consequence we answer a question raised by Pelant, Holický and Kalenda. We also discuss a few results and conjectures concerning a generalization of this problem.