Colourings avoiding symmetrically coloured arithmetic progressions
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How few colors suffice to color {1,…,N} with no nontrivial k-term arithmetic progression whose colors agree symmetrically from its two ends? For four terms the target improves the known power exponent toward subpolynomial growth.
Why it matters and what counts as progress
Why it matters: Better colorings quantify how efficiently symmetric progression patterns can be avoided.
What counts as progress: A smaller color-count exponent, including a proved O(N^(1/4)) four-term construction, or further progress toward N^o(1); those two targets are distinct.
How the problem developed
- 2023 · Baseline construction [1]
Mingyang Deng, Jonathan Tidor and Yufei Zhao Gave an O(N^(log₂₂3))-colouring of [N] with no nontrivial symmetrically coloured 4-term progression.
- 13 Jul 2023 · Problem formulation [1]
Mingyang Deng, Jonathan Tidor and Yufei Zhao Connected colorings avoiding symmetrically coloured progressions to a question about Fourier-uniform sets and 4-term arithmetic progressions.
- 22 Jul 2026 · AI-assisted construction [2]
Ruizhe Shi and Yiqi Dong Improved the 4-term exponent to 1/4 and gave Oₖ(N^(4/k²)) colourings for even k≥4, disproving Gowers’ proposed lower bound for k≥6.
AI contribution
An Improved Upper Bound for Colorings Without Symmetrically Colored k-Term Arithmetic Progressions
For every even k≥4 and prime p>k, a coloring of Z/(p^(k²/4))Z uses (k−1)^(k²/4−1)p colors and avoids nontrivial symmetrically colored k-term arithmetic progressions. At k=4 this gives 27p colors and an O(N^(1/4)) interval bound.
Sources
- Deng, Tidor and Zhao · Uniform sets with few progressions via colorings (2023; published 2025)
- Shi and Dong · An improved upper bound for colorings without symmetrically coloured k-term arithmetic progressions (2026)
- An Improved Upper Bound for Colorings Without Symmetrically Colored k-Term Arithmetic Progressions
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