Computing majority via multiple queries

We study the following combinatorial problem. Let B   be the set of balls such that each ball is in one of two colors, and let k≥2k≥2 be an integer. A query consists of selecting k balls from the set B and the answer to the query is either that (i) all balls have the same color, or (ii) there exist two balls of different colors and a pair of those is pointed out. The goal is to design a strategy that uses the minimum number of queries that allows to produce a ball of the majority color, or to determine that both colors are equally represented in B  . The solution is known in cases k=2k=2 and k=3k=3 and lower bounds for the minimum number of queries has been given for every k  . In this paper we give exact numbers for every k≥3k≥3. The tool is to build a bipartite graph over B with pairs of different color balls as edges.