Question 42
Given positive integers $m \ge 2$, $n \ge 2$, consider the set
$$
S(m, n) = \left\{ \mathbf{x} = (x_1, x_2, \dots, x_m) ; \mathbf{x} \in \mathbb{Z}_+^m, \sum_{k=1}^m x_k = n \right\}.
$$
Determine $N(m, n)$, the largest cardinality $|X|$ of a subset $X \subseteq S(m, n)$ with
$$
|X| = |\{x_k ; \mathbf{x} \in X\}|, \text{ for all } k = 1, 2, \dots, m.
$$
Vasile Pop & Dan Schwarz
$$
S(m, n) = \left\{ \mathbf{x} = (x_1, x_2, \dots, x_m) ; \mathbf{x} \in \mathbb{Z}_+^m, \sum_{k=1}^m x_k = n \right\}.
$$
Determine $N(m, n)$, the largest cardinality $|X|$ of a subset $X \subseteq S(m, n)$ with
$$
|X| = |\{x_k ; \mathbf{x} \in X\}|, \text{ for all } k = 1, 2, \dots, m.
$$
Vasile Pop & Dan Schwarz