On some maximal and minimal sets

Abstract

A set A of positive integers is called 3-free if it contains no 3-term arithmetic progression. Furthermore, such A is called maximal if it is not properly contained in any other 3-free set. In 2006, by confirming a question posed by Erdős et al., Savchev and Chen proved that there exists a maximal 3-free set \(\{a_1<a_2<\cdots<a_n<\cdots \}\) of positive integers with the property that \(\lim _{n\rightarrow \infty }(a_{n+1}-a_n)=\infty \). In this paper, we generalize their result. On the other hand, a set A of nonnegative integers is called an asymptotic basis of order h if every sufficiently large integer can be represented as a sum of h elements of A. Such A is defined as minimal if no proper subset of A has this property. We also extend a result of Jańczak and Schoen about the minimal asymptotic basis.