Avoiding four-term progressions from finitely many starts
Abstract
For every finite set A⊂ℤ, we construct a bijection p:ℕ0→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≥2 the enumeration can begin with 0,1,3,…,2n−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.
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BibTeX
@misc{thompson2026avoiding,
title = {{Avoiding four-term progressions from finitely many starts}},
author = {William Thompson},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13350},
eprint = {2609.13350},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13350},
}
RIS
TY - UNPB
TI - Avoiding four-term progressions from finitely many starts
AU - William Thompson
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.13350
UR - https://arxiv.org/abs/2609.13350
AB - For every finite set A⊂ℤ, we construct a bijection p:ℕ_0→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≥2 the enumeration can begin with 0,1,3,…,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.
ER -