Upper Bounds For Families Without Weak Delta-Systems

Abstract

For $k\ge 3$, a collection of $k$ sets is said to form a \emph{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 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 $\Theta (C)$ is the capset capacity. In particular, this shows that \[ |\mathcal{F}|\leq(1.8367\dots+o(1))^{n}. \]

6 pages. Minor changes