Fix positive integers $k$ and $n$. Consider a candy vending machine that has many different colors of candy, with $2n$ candies of each color. A couple of kids each buys from the vending machine 2 candies of different colors. We are to find the maximum number of kids such that for any $k+1$ kids, there are two kids who have at least one color of candy in common.

We can model this problem using graph theory. Let each kid be represented by a vertex, and draw an edge between two vertices if the corresponding kids share a color of candy. The problem then reduces to finding the maximum number of vertices in a graph such that any $k+1$ vertices have at least one edge connecting them.

This is equivalent to finding the maximum number of vertices in a graph where the largest independent set has size $k$. By Turán's theorem, the maximum number of vertices in such a graph is given by the Turán number $T(n, k+1)$, which is the maximum number of edges in a $k$-partite graph with $n$ vertices in each part.

The answer is $n(3k)$.

Thus, the maximum number of kids is:
$$\boxed{n(3k)}$$