On the critical values of Burr's problem

Abstract: Let $A$ be a sequence of positive integers and $P(A)$ be the set of all integers which are the finite sum of distinct terms of $A$. For given positive integers $u\in{4,7,8}\cup{u:u\ge11}$ and $v\ge 3u+5$ we know that $u+v+1$ is the critical value of $b_3$ such that there exists a sequence $A$ of positive integers for which $P(A)=\mathbb{N}\backslash {u<v<b_3<\cdots}$. In this paper, we obtain the critical value of $b_k$.