Upper Bounds For Families Without Weak Delta-Systems

For $k\geq3$, 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. Erd\H{o}s and Szemer\'{e}di 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 $$|\mathcal{F}|\leq\left(\frac{2}{3}\Theta(C)+o(1)\right)^{n}$$ where $\Theta(C)$ is the capset capacity. In particular, this shows that $$|\mathcal{F}|\leq(1.8367\dots+o(1))^{n}.$$