Let be a set of -dimensional row vectors with entries in a -ary alphabet. A matrix with entries in the same -ary alphabet is -constrained if every set of columns of contains as a submatrix a copy of the vectors in , up to permutation. For a given set of -dimensional vectors, we compute the asymptotic probability for a random matrix to be -constrained, as the numbers of rows and columns both tend to infinity. If is the number of columns and the number of rows, then the threshold is at , where only depends on the dimension of vectors and not on the particular set . Applications to superimposed codes, shattering classes of functions, and Sidon families of sets are proposed. For , an explicit construction of a -constrained matrix is given.