AbstractWe study the maximum size and the structure of sets of natural numbers which contain no solution of one or more linear equations. Thus, for every natural i and k⩾2, we find the minimum α=α(i,k) such that if the upper density of a strongly k-sum-free set A⊆N is at least α, then A is contained in a maximal strongly k-sum-free set which is a union of at most i arithmetic progressions. We also determine the maximum density of sets of natural numbers without solutions to the equation x=y+az, where a is a fixed integer