A Generalisation of Maximal (k,b)-Linear-Free Sets of Integers

Fix integers k, b, q with k ≥ 2, b ≥ 0, q ≥ 2. Define the function p to be: p(x) = kx+b. We call a set S of integers (k, b, q)-linear-free if x ∈ S implies pi(x) ∉ S for all i = 1, 2, . . . , q - 1, where pi(x) = p ( pi-1(x) ) and p0(x) = x. Such a set S is maximal in [n] := {1, 2, . . . , n} if S ∪ {t}, ∀t ∈ [n]S is not (k, b, q)-linear-free. Let Mk,b,q(n) be the set of all maximal (k, b, q)-linear-free subsets of [n], and define gk,b,q(n) = minS ∈Mk,b,q(n) |S | and fk,b,q(n) = maxS ∈Mk,b,q(n) |S |. In this paper, formulae for gk,b,q(n) and fk,b,q(n) are proposed. Also, it is proven that there is at least one maximal (k, b, q)-linear-free subset of [n] with exactly x elements for any integer x between gk,b,q(n) and fk,b,q(n), inclusively. © 2024 the Author(s).