The largest $(k,\ell )$-sum-free subsets

Abstract

Let ${\mathcal{M}}_{(2,1)}(N)$ be the infimum of the largest sum-free subset of any set of $N$ positive integers. An old conjecture in additive combinatorics asserts that there is a constant $c=c(2,1)$ and a function $\omega (N)\to \infty$ as $N\to \infty$, such that $cN+\omega (N)<{\mathcal{M}}_{(2,1)}(N)<(c+o(1))N$. The constant $c(2,1)$ is determined by Eberhard, Green, and Manners, while the existence of $\omega (N)$ is still wide open. In this paper, we study the analogous conjecture on $(k,\ell )$-sum-free sets and restricted $(k,\ell )$-sum-free sets. We determine the constant $c(k,\ell )$ for every $(k,\ell )$-sum-free sets, and confirm the conjecture for infinitely many $(k,\ell )$.

33 pages; accepted for publication in Trans. Amer. Math. Soc