The Maximum Number of Subset Divisors of a Given Size

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 $\left(\genfrac{}{}{0ex}{}{n-1}{k}\right)$ 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 $\left(\genfrac{}{}{0ex}{}{n-1}{k}\right)$ with only finitely many exceptions for each $s$.

submitted on July 17, 2014