Upper Bounds For Families Without Weak Delta-Systems

TL;DR

This paper improves upper bounds on the size of families of subsets of {1,{...},{n}} that do not contain weak -systems, connecting combinatorial set theory with capset capacity to refine previous results.

Contribution

The paper provides a tighter upper bound on the maximum size of families avoiding weak -systems, utilizing capset capacity to advance prior bounds.

Findings

Established an upper bound of approximately (1.8367+o(1))^n for such families.

Connected combinatorial bounds with capset capacity to improve existing results.

Enhanced the understanding of the structure of families avoiding weak -systems.

Abstract

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 \[ |\mathcal{F}|\leq(1.8367\dots+o(1))^{n}. \]