Consider the corners of the (say) unit n-cube: {0,1}^n, in R^n. QUESTION: What is the size |X(n)| of a largest subset X(n) of the 2^n corners, such that all nonzero distances in X(n) have the same value? The distinct distances are {sqrt(k) | 0 <= k <= n}. For a fixed k > 0, the number of distances = sqrt(k) is 2^(n-1) * C(n,k), where C(n,k) := n!/(k! (n-k)!). MORE GENERAL QUESTION: What is the size |X(n,k)| of a largest subset X(k,n) having all its nonzero distances = sqrt(k) ? (These questions can be rephrased to ask what is the size of the largest clique in the constant-distance graphs for {0,1}^n.) F'rinstance, in {0,1}^4, the graph of all distances = sqrt(2) has two components. Each is isomorphic to what you get if you connect all pairs of vertices of an octagon (or 3-cube), except antipodes, with an edge. So each component of the graph has 8 vertices and 24 edges. The largest clique has only 4 vertices. --Dan