\nu_0(G)$ and prove this conjecture in Theorem \[cycle\] for any cycle.

Monomial ideals with index $>1$.
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Let $K$ be a field, $S=K[x_1,\ldots,x_n]$ the polynomial ring over $K$ in $n$ indeterminates, and let $I\subset S$ be a monomial ideal generated in degree $d$.

The ideal is called [*$r$ steps linear*]{}, if $I$ has a linear resolution up to homological degree $r$, in other words, if $\beta_{i,i+j}(I)=0$ for all pairs $(i,j)$ with $0\leq i\leq r$ and $j>d$. Then the number $$\index(I)=\sup\{r\: \text{$I$ is $r$ steps linear}\}+1$$ is called the [*index*]{} of $I$. In particular, $I$ has a linear resolution if and only if $\index(I)=\infty$, and $I$ is linearly related if and only if $\index(I)>1$.

In this section we derive a criterion for a monomial ideal to be linearly related. This criterion is actually an immediate consequence of the lcm-lattice formula [@GPW] by Gasharov, Peeva and Welker for the multi-graded Betti numbers of a monomial ideal $I$, not necessarily generated in a single degree.

Let $G(I) = \{u_1,\ldots,u_m\}$ be the unique minimal set of monomial generators of $I$. We denote by $L(I)$ the lcm-lattice of $I$, i.e. the poset whose elements are labeled by the least common multiples of subsets of monomials 