# A bound on partitioning clusters

## TL;DR

This paper establishes a new upper bound on the number of cluster partitions in a finite set collection, improving previous bounds and applying elementary calculus and numerical methods, with implications for phylogenetic algorithms.

## Contribution

It introduces a novel bound on cluster partition counts and demonstrates the optimality of the exponent, advancing understanding in combinatorial bounds related to phylogenetic analysis.

## Key findings

- Bound on cluster partition count: |X|^{3/p} with p ≈ 1.73814
- Improved upon the trivial quadratic bound |X|^2
- Optimal exponent for additive energy in discrete cubes: |A|^{log_2 6}

## Abstract

Let $X$ be a finite collection of sets (or "clusters"). We consider the problem of counting the number of ways a cluster $A \in X$ can be partitioned into two disjoint clusters $A_1, A_2 \in X$, thus $A = A_1 \uplus A_2$ is the disjoint union of $A_1$ and $A_2$; this problem arises in the run time analysis of the ASTRAL algorithm in phylogenetic reconstruction. We obtain the bound $$ | \{ (A_1,A_2,A) \in X \times X \times X: A = A_1 \uplus A_2 \} | \leq |X|^{3/p} $$ where $|X|$ denotes the cardinality of $X$, and $p := \log_3 \frac{27}{4} = 1.73814\dots$, so that $\frac{3}{p} = 1.72598\dots$. Furthermore, the exponent $p$ cannot be replaced by any larger quantity. This improves upon the trivial bound of $|X|^2$. The argument relies on establishing a one-dimensional convolution inequality that can be established by elementary calculus combined with some numerical verification. In a similar vein, we show that for any subset $A$ of a discrete cube $\{0,1\}^n$, the additive energy of $A$ (the number of quadruples $(a_1,a_2,a_3,a_4)$ in $A^4$ with $a_1+a_2=a_3+a_4$) is at most $|A|^{\log_2 6}$, and that this exponent is best possible.