The Maximum Number of Subset Divisors of a Given Size

TL;DR

This paper investigates the maximum number of k-subsets of an n-element set that can be s-divisors, disproves a conjecture for s=1, and establishes conditions under which the maximum is binomial coefficient with finitely many exceptions.

Contribution

It provides a counterexample to Huynh's conjecture for s=1 and proves that adding a necessary condition yields the maximum number as a binomial coefficient with finitely many exceptions.

Findings

Counterexample to Huynh's conjecture for s=1

Maximum number of s-divisors is binomial coefficient under certain conditions

Finite exceptions identified for the maximum number of s-divisors

Abstract

If $s$ is a positive integer and $A$ is a set of positive integers, we say that $B$ is an $s$-divisor of $A$ if $\sum_{b\in B} b\mid s\sum_{a\in A} a$. We study the maximal number of $k$-subsets of an $n$-element set that can be $s$-divisors. We provide a counterexample to a conjecture of Huynh that for $s=1$, the answer is $\binom{n-1}{k}$ with only finitely many exceptions, but prove that adding a necessary condition makes this true. Moreover, we show that under a similar condition, the answer is $\binom{n-1}{k}$ with only finitely many exceptions for each $s$.