Maximum k-sum n-free sets of the 2-dimensional integer lattice

Abstract

For a positive integer n, let [n] denote {1, …, n}. For a 2-dimensional integer lattice point b and positive integers k ≥ 2 and n, a k-sum b-free set of [n] × [n] is a subset S of [n] × [n] such that there are no elements a1, …, ak in S satisfying a1 + · · · + ak = b. For a 2-dimensional integer lattice point b and positive integers k ≥ 2 and n, we determine the maximum density of a k-sum b-free set of [n] × [n]. This is the first investigation of the non-homogeneous sum-free set problem in higher dimensions.