generated in degree $3$ with linear resolution, that is, $\index(I)=\infty$, and with the property that $\index(I^2)=1$.

Theorem \[main\] implies in particular that for a monomial ideal generated in degree $2$ we have $\index(I)=1$ if and only if $\index(I^k)=1$ for all $k$ . Again this fails if $I$ is not generated in degree $2$. Indeed, for $n\geq 4$ consider the ideal $I=(x^n,x^{n-1}y,y^{n-1}x,y^n)$. Then $\index(I^k)=1$ for $k=1,\ldots,n-3$ and $\index(I ^k)=\infty$ for $k>n-3$. There are also many such counterexamples of monomial ideals generated in degree $3$.

The ideal $I$ in the example of Nevo and Peeva has index 2, its square has index 7, while $I^3$ and $I^4$ have a linear resolution. This example and other experimental evidence leads us to conjecture that for a monomial ideal $I$ generated in degree $2$, one has $\index(I^{k+1})>\index(I^k)$ for all $k$ if $\index(I)>1$. This conjecture implies that $\index(I^k)>k$ if $\index(I)>1$. In particular, if $G$ is a graph (possibly with loops) on the vertex set $[n]$, this would imply that $I(G)^k$ has a linear resolution for $k>n-2$.

For the proof of our Theorem \[main\] we use the theory of $\lcm$-lattices introduced by Gasharov, Peeva and Welker [@GPW]. As an easy application of their theory the monomial ideals 