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Paper:

TR24-193 | 22nd November 2024 20:53

Reasonable Bounds for Combinatorial Lines of Length Three

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TR24-193
Authors: Amey Bhangale, Subhash Khot, Yang P. Liu, Dor Minzer
Publication: 22nd November 2024 20:58
Downloads: 410
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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 $3^{-n}|A| \ge \Omega((\log^* n)^{-1/2})$ of [D.H.J. Polymath, Ann. of Math. 2012].



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