On deleting coordinates from integer vectors Original Research Article

In many vectorial settings, the trivial linear lower bound on the size of the shadow is asymptotically best possible. To be more precise, let T(d) be the set of vectors of dimension d over {1,2,…,g}. As usual, for a subset S of T(d), we obtain its shadow, ΔS, as a subset of T(d−1), by deleting coordinates of vectors in S in all possible ways. Further, let f(N) be the minimum of |ΔS| taken over all subsets S of T(d) such that |S|=N. Then our main result is that the graph of f(N) converges to a straight line, as d→∞. As a corollary, the analogous results hold for matrices, circles, triangles, cubes, pyramids, and the like.