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 ) = k x + b . We call a set S of integers ( k , b , q ) -linear-free if x ∈ S implies p i ( x ) ∉ S for all i = 1 , 2 , … , q − 1 , where p i ( x ) = p ( p i − 1 ( x ) ) and p 0 ( 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 M k , b , q ( n ) be the set of all maximal ( k , b , q ) -linear-free subsets of [ n ] , and define g k , b , q ( n ) = min S ∈ M k , b , q ( n ) | S | and f k , b , q ( n ) = max S ∈ M k , b , q ( n ) | S | . In this paper, formulae for g k , b , q ( n ) and f k , 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 g k , b , q ( n ) and f k , b , q ( n ) , inclusively.