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

Let 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 ω(N) → ∞ as N → ∞, such that cN + ω(N) < M(2,1)(N) < (c + o(1))N. The constant c(2, 1) is determined by Eberhard, Green, and Manners, while the existence of ω(N) is still wide open. In this paper, we study the analogous conjecture on (k,ℓ)-sum-free sets and restricted (k,ℓ)-sum-free sets. We determine the constant c(k,ℓ) for every (k,ℓ)-sum-free sets, and confirm the conjecture for infinitely many (k,ℓ).