Moment obstructions and continuum-to-discrete bounds for checkerboard no-three-in-line sets

Abstract

Fix one colour class in the checkerboard colouring of an n× n integer grid, and let M₄(n,ε) be the largest subset having at most two points in every row, column, and diagonal of slopes ±1. We prove the near-saturation bound M₄(n,ε)≤2n−4 for n≥6. The proof uses first and second moments of the four line families: a hypothetical set of size 2n−3 produces row, column, and diagonal deficits whose exact moment identities contradict Cauchy--Schwarz. A finite argument handles n=6. The same identity extends to arbitrary deficit multisets. It gives M₄(n,ε)≤2n−d whenever d≥4 is an integer and n≥3d−4, and an entirely discrete asymptotic estimate M₄(n,ε)≤(√(21)−3)n+8. We also prove a general continuum-to-discrete theorem for the associated four-direction fractional packing problem. Applying it to the exact continuum dual certificate constructed in earlier work yields, for both colours, L_mono(n,ε)≤αn+O(1), α≈1.5768233968738, and hence the same upper bound for M₄ and for checkerboard no-three-in-line sets.

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BibTeX RIS

BibTeX

@misc{aujla2026moment,
  title = {{Moment obstructions and continuum-to-discrete bounds for checkerboard no-three-in-line sets}},
  author = {Jujhar Aujla and Thomas Prellberg and Nirvair Sandhu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.12340},
  eprint = {2609.12340},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.12340},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Moment obstructions and continuum-to-discrete bounds for checkerboard no-three-in-line sets
AU  - Jujhar Aujla
AU  - Thomas Prellberg
AU  - Nirvair Sandhu
PY  - 2026
DA  - 2026/09/19
PB  - arXiv
DO  - 10.48550/arxiv.2609.12340
UR  - https://arxiv.org/abs/2609.12340
AB  - Fix one colour class in the checkerboard colouring of an n× n integer grid, and let M_4(n,ε) be the largest subset having at most two points in every row, column, and diagonal of slopes ±1. We prove the near-saturation bound M_4(n,ε)≤2n−4 for n≥6. The proof uses first and second moments of the four line families: a hypothetical set of size 2n−3 produces row, column, and diagonal deficits whose exact moment identities contradict Cauchy--Schwarz. A finite argument handles n=6. The same identity extends to arbitrary deficit multisets. It gives M_4(n,ε)≤2n−d whenever d≥4 is an integer and n≥3d−4, and an entirely discrete asymptotic estimate M_4(n,ε)≤(√(21)−3)n+8. We also prove a general continuum-to-discrete theorem for the associated four-direction fractional packing problem. Applying it to the exact continuum dual certificate constructed in earlier work yields, for both colours, L_mono(n,ε)≤αn+O(1), α≈1.5768233968738, and hence the same upper bound for M_4 and for checkerboard no-three-in-line sets.
ER  -

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