Extremal Problems for Subset Divisors

TL;DR

This paper determines the maximum number of divisor subsets for any set of n positive integers and characterizes all extremal sets, also exploring a related k-subset problem.

Contribution

It provides exact solutions for the maximum number of divisor subsets for all n and characterizes the extremal sets, including a special case for the k-subset variant.

Findings

Exact maximum number of divisor subsets for all n

Characterization of extremal sets achieving the maximum

Results for the k-subset analogue when n=2k

Abstract

Let $A$ be a set of $n$ positive integers. We say that a subset $B$ of $A$ is a divisor of $A$, if the sum of the elements in $B$ divides the sum of the elements in $A$. We are interested in the following extremal problem. For each $n$, what is the maximum number of divisors a set of $n$ positive integers can have? We determine this function exactly for all values of $n$. Moreover, for each $n$ we characterize all sets that achieve the maximum. We also prove results for the $k$-subset analogue of our problem. For this variant, we determine the function exactly in the special case that $n=2k$. We also characterize all sets that achieve this bound when $n=2k$.