Let $X_i$ be the family of multisets defined as $X_i:=\lbrace\lbrace 1^{i_1}, \dots, n^{i_n}\rbrace| i_j \leq i \text{ for every } j=1,\dots, n\rbrace$. Let $X:=\bigcup_{i=1}^{\infty} X_i$.
Let $A$ be the multiset $A=\lbrace 1^{a_1}, \dots, n^{a_n} \rbrace \in X$. We define the cardinality of $A$ as $|A|:=\sum_{i=1}^n a_i$.
Let $X^d$ be the subset of $X$ containing all the multisets of cardinality $d$.
Let $\mathcal{A}:=\lbrace A_1, \dots, A_k \rbrace$ be a family of $k$ elements in $X^d$ for a given $d$. Let $D>d$; we define $\mathcal{A}^D$ to be the subset of $X^D$ defined as: $$\mathcal{A}^D:= \bigcup_{i=1}^k \lbrace I \in X^D | A_i \subseteq I \rbrace.$$
My goal is to find the choice of $\mathcal{A}$ that minimizes $|\mathcal{A}^D|$.
My intuition suggests me that (up to permutation of the objects $1, \dots, n$), the optimal choice for $\mathcal{A}$ should be the first $k$ elements in the lexicographic order. e.g. for $n=3$, $d=4$, $k=5$ they would be: $$\mathcal{A}=\lbrace \lbrace 1^4 \rbrace, \lbrace 1^3,2 \rbrace, \lbrace 1^3,3 \rbrace, \lbrace 1^2,2^2 \rbrace, \lbrace 1^2,2,3 \rbrace \rbrace.$$
Many experiments has corroborated this conjecture.
This can be rephrased in the following way: find the set of size $k$ of monomials of degree $d$ in $\mathbb{F}[x_1, \dots, x_n]$, that minimizes the number of monomials of degree $D$, that are multiples of at least one of the monomials in the set.
I managed to find a solution for the polynomial case, when one considers the quotient algebra $\mathbb{F}[x_1, \dots, x_n]/(x_i^2)$, but it appeared pretty complicated for the simplicity of the statement.
Again, for the simplicity of the statement, I believe that someone has already tackled (and probably solved) the problem. Unfortunately, I do not have a background in combinatorics and I do not know how I should formulate this problem properly in order to find the studies related to it. Any help in this direction will be deeply appreciated.
After searching for a while, I could find that, $\mathcal{A}$ is the set containing the first $k$ elements according to the lexicographic order. This statement is the content of Corollary 1 of the paper by Clements and Lindström, titled "A generalization of a combinatorial theorem of macaulay" with doi: https://doi.org/10.1016/S0021-9800(69)80016-5