Let $I$ and $J$ be two monomial ideals in some polynomial ring $S=k[X_1,...,X_n]$. Furthermore, assume that $G(I)\subseteq G(J)$, where $G(I)$ denotes the minimal set of monomial generators of $I$, and likewise for $J$. Is it true that $\beta_{ij}(I)\le \beta_{ij}(J)$ for all $i,j$? This is certainly the case for $i=0$, since $\beta_{0j}$ is the number of elements in the minimal set of generators of degree $j$. I was also wondering if
$$\min\{j:\beta_{ij}(I)\neq 0\}\le \min\{j:\beta_{ij}(J)\neq 0\}$$
which would actually be a consequence of the previous statement if it were correct.