Geometry of coin-weighing problems

Given a set of m coins out of a collection of coins of k unknown distinct weights, we wish to decide if all the m given coins have the same weight or not using the minimum possible number of weighings in a regular balance beam. Let m(n, k) denote the maximum possible number of coins for which the above problem can be solved in n weighings. We show that m(n, 2) = n^{( 1/2 +o(l))n}, whereas for all 3≤k≤n+1, m(n, k) is much smaller than m(n, 2) and satisfies m(n, k) = Θ(n log n/log k). The proofs have an interesting geometric flavour, and combine Linear Algebra techniques with geometric, probabilistic and combinatorial arguments.