TITLE: Can we rearrange the integers, such that the arithmetic average of any two numbers does not appear between them?
QUESTION [8 upvotes]: Can we rearrange the integers, such that the arithmetic average of any two numbers does not appear between them?
In other words: Can we have a sequence $\{a_n\}_{n\in \mathbb{Z}}$, where all integers appear once and only once, such that $a_j\ne (a_i+a_k)/2,\forall i<j<k$?

I've noticed the questions here and here. But the first one only applies to arbitrarily long finite segments. The second does not contain a solution in itself, and unfortunately I can not access the reference (for free).
Also, the second question is much more general than this one, so I am wondering if this one can be solved somewhat more easily.

REPLY [2 votes]: Collecting references from the comments: The paper On permutations containing no long arithmetic progressions (by Davis, Entringer, Graham and Simmons in Acta Aritmetica 34(1), 1977) shows that this is not possible (Fact 5 on page 85).
Actually what is shown is that a doubly-infinite list of naturals can't have this property, but if you have a a doubly-infinite list of integers and delete the negative numbers, you either get that or a singly infinite list of naturals, for which it is easy to see that the property is impossible.