Candidates for non-zero Betti numbers of monomial ideals

Let $I$ be a monomial ideal in the polynomial ring $S$ generated by elements of degree at most $d$. In this paper, it is shown that, if the $i$-th syzygy of $I$ has no element of degrees $j,\dots ,j+(d-1)$ (where $j\ge i+d$), then $(i+1)$-syzygy of $I$ does not have any element of degree $j+d$. Then we give several applications of this result, including an alternative proof for Green-Lazarsfeld index of the edge ideals of graphs as well as an alternative proof for Fr"oberg’s theorem on classification of square-free monomial ideals generated in degree two with linear resolution. Among all, we describe the possible indices $i,j$ for which $I$ may have non-zero Betti numbers ${\beta }_{i,j}$.