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Source. Part 3 (Open problems), Problem 4, p. 8, of P. Erdős and M. B. Nathanson, "Partitions of bases into disjoint unions of bases," J. Number Theory 29 (1988), no. 1, 1--9. The edition read is identified on the source card.

Statement

Definition (p. 8). An asymptotic basis $A$ of order $h$ is minimal if no proper subset of $A$ is an asymptotic basis of order $h$.

Problem 4 (p. 8). The paper says it is not clear whether there is a relationship between asymptotic bases that contain minimal asymptotic bases and asymptotic bases that are disjoint unions of two asymptotic bases. As an example it asks: let $A_1$ and $A_2$ be asymptotic bases of order 2 with $A_1\cap A_2=\emptyset$, and let $A=A_1\cup A_2$. Does $A$ contain a minimal asymptotic basis of order 2?

The paper gives no result on the question.

Read depth. Claims checked: the problem was read clause by clause on the print.

Dependencies

None.

Bears on

• #869: the problem page lists this paper among its references; its question is the one Problem 4 asks. The paper poses the question and does not answer it.