# Convolution estimates and the number of disjoint partitions

## TL;DR

This paper provides an upper bound estimate for the number of disjoint partitions within a finite set collection, extending previous results to arbitrary numbers of partitions using convolution estimates.

## Contribution

It generalizes the Kane-Tao result for three partitions to any finite number of disjoint partitions with a new bounding function.

## Key findings

- Derived an explicit upper bound for the count of disjoint partitions.
- Extended previous specific case to general finite cases.
- Provided a mathematical estimate involving the function c(n).

## Abstract

Let $X$ be a finite collection of sets. We count the number of ways a disjoint union of $n-1$ subsets in $X$ is a set in $X$, and estimate this number from above by $|X|^{c(n)}$ where $$ c(n)=\left(1-\frac{(n-1)\ln (n-1)}{n\ln n} \right)^{-1}. $$ This extends the recent result of Kane-Tao, corresponding to the case $n=3$ where $c(3)\approx 1.725$, to an arbitrary finite number of disjoint $n-1$ partitions.