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arXiv · 2411.15137

Reasonable Bounds for Combinatorial Lines of Length Three

Amey Bhangale·Subhash Khot·Yang P. Liu·Dor Minzer
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Abstract

We prove that any subset $A \subseteq [3]^n$ with $3^{-n}|A| \ge (\log\log\log\log n)^{-c}$ contains a combinatorial line of length $3$, i.e., $x, y, z \in A$, not all equal, with $x_i=y_i=z_i$ or $(x_i,y_i,z_i)=(0,1,2)$ for all $i = 1, 2, \dots, n$. This improves on the previous best bound of $3^{-n}|A| \ge Ω((\log^* n)^{-1/2})$ of [D.H.J. Polymath, Ann. of Math. 2012].

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Amey Bhangale, Subhash Khot, Yang P. Liu, Dor Minzer. 2024-11-22. Reasonable Bounds for Combinatorial Lines of Length Three. https://arxiv.org/abs/2411.15137

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