Avoiding four-term progressions from finitely many starts

Abstract

For every finite set $A\subset\mathbb{Z}$, we construct a bijection $p:\mathbb{N}_0\to\mathbb{Z}$ beginning with $0,1$ that contains no four-term arithmetic progression, in occurrence order, whose first value belongs to $A$. More generally, for each $n\ge2$ the enumeration can begin with $0,1,3,\ldots,2^{n-1}-1$ and simultaneously avoid every such progression starting in $A$ or among these first $n$ entries. The construction uses finitely branching prerequisite relations and an explicit bounded integer potential. This proves finite prerequisite closure and gives an exhaustive enumeration, rather than merely a total order. We also give an extension theorem for finite prefixes with compatible binary-tail constraints. These results do not determine whether every enumeration of the integers contains an ordered four-term arithmetic progression.