Colorful flowers

For a set A let [ A ] k denote the family of all k-element subsets of A. A function f : [ A ] k → C is a local coloring if it maps disjoint sets of A into different elements of C. A family F ⊆ [ A ] k 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 [ A ] k 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.