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Upper Bounds For Families Without Weak Delta-Systems

Combinatorica volume 43, pages 729–735 (2023)Cite this article

Abstract

For \(k\ge 3\), a collection of k sets is said to form a weak \(\Delta \)-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 \(\Delta \)-system. In this note we improve upon the best upper bound due to the author and Sawin, and show that

$$\begin{aligned} |\mathcal {F}|\le \left( \frac{2}{3}\Theta (C)+o(1)\right) ^{n} \end{aligned}$$

where \(\Theta (C)\) is the capset capacity. In particular, this shows that

$$\begin{aligned} |\mathcal {F}|\le (1.8367\ldots +o(1))^{n}. \end{aligned}$$

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Acknowledgements

I would like to thank Lisa Sauermann for her many helpful comments, and for pointing out an error in the original version of this paper. I would also like to thank the anonymous referees for their valuable feedback.

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    Eric Naslund

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Naslund, E. Upper Bounds For Families Without Weak Delta-Systems. Combinatorica 43, 729–735 (2023). https://doi.org/10.1007/s00493-023-00032-9

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