## 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

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

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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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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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DOI: https://doi.org/10.1007/s00493-023-00032-9