The European zoos with at least two types of species are separated in two groups $\hat{A}$ and $\hat{B}$ in such a way that every pair of zoos $(A,B)$ $(A\in\hat{A}, B\in\hat{B})$ have some animal in common. What is the least $k$ for which we can color the cages in the zoos (each cage only has all animals of one species) such that no zoo has cages of only one color (with every animal across all zoos having to be colored in the same color)? For the maximal value of $k$, find all possibilities (zoos and species), for which this maximum is achieved.