Some results on the subadditivity condition of syzygies

Among other results, we prove that if $I$ is a monomial ideal of $S=K[x_1,\dots ,x_n]$, where $K$ is a field, and $a\ge b-1\ge 0$ are integers such that $a+b\le \mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}\mathrm{d}\mathrm{i}\mathrm{m}(S\mathrm{/}I)$, then $$t_{a+b}\le t_a+t_1+t_2+\cdots +t_b-\frac{b(b-1)}{2},$$ where $t_1,t_2,\dots$ are the maximal shifts in the minimal graded free $S$-resolution of $S\mathrm{/}I$.