Threshold functions for systems of equations on random sets

We present a unified framework to deal with threshold functions for the existence of certain combinatorial structures in random sets. More precisely, let M · x = 0 be a linear system of r equations and m variables, and A a random set on [n] where each element is chosen independently with the same probability. We show that, under certain conditions, there exists a threshold function for the property “Am contains a non-trivial solution of M · x = 0”, depending only on r and m and, furthermore, we study the behavior of the limiting probability in the threshold scale in terms of volumes of certain convex polytopes arising from the linear system under study. Our results cover several combinatorial families, namely sets without arithmetic progressions of given length, k-sum-free sets, Bh[g] sequences and sets without Hilbert cubes of dimension k, among others.