Extremal problems for subset divisors

Tony Huynh¹

¹University of Rome

The electronic journal of combinatorics, Tome 21 (2014) no. 1

Résumé

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 $n=2k$.

Détail

DOI : 10.37236/3438

Classification : 05D05, 05A15
Mots-clés : extremal combinatorics, exact enumeration