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arXiv · 2203.13370

Upper Bounds For Families Without Weak Delta-Systems

Eric Naslund
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Abstract

For $k\geq3$, a collection of $k$ sets is said to form a \emph{weak $Δ$-system} if the intersection of any two sets from the collection has the same size. Erdős and Szemerédi asked about the size of the largest family $\mathcal{F}$ of subsets of $\{1,\dots,n\}$ that does not contain a weak $Δ$-system. In this note we improve upon the best upper bound of the author and Sawin from arXiv:1606.09575 and show that \[ |\mathcal{F}|\leq\left(\frac{2}{3}Θ(C)+o(1)\right)^{n} \] where $Θ(C)$ is the capset capacity. In particular, this shows that \[ |\mathcal{F}|\leq(1.8367\dots+o(1))^{n}. \]

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Eric Naslund. 2023-01-22. Upper Bounds For Families Without Weak Delta-Systems. https://arxiv.org/abs/2203.13370

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