assumption that $G$ is gap and cricket free. Here we should note that claw free implies cricket free.

In this paper we attempt to generalize the result of Hibi, Zheng and the second author of this paper in a different direction. An ideal $I$ is called $r$ steps linear, if $I$ has a linear resolution up to homological degree $r$. In other words, if $I$ is generated in a single degree, say $d$, and $\beta_{i,i+j}(I)=0$ for all pairs $(i,j)$ with $0\leq i\leq r$ and $j>d$. The number $$\index(I)=\sup\{r\: \text{$I$ is $r$ steps linear}\}+1$$ is called the index of $I$. The main result of Section 2 (Theorem \[main\]) is the following: Let $I$ be a monomial ideal generated in degree $2$. We interpret $I$ as the edge ideal of a graph $G$ which may also have loops (corresponding to squares among the monomial generators of $I$). Then the following conditions are equivalent: $G$ is gap free, i.e. no induced subgraph of $G$ consists of two disjoint edges, (b) $\index(I^k)>1$ for all $k$, (c) $\index(I^k)>1$ for some $k$.

Theorem \[main\] is not valid for monomial ideals generated in degree $>2$. There is an example by Conca [@Co] of a monomial ideal $I$ 