squarefree powers of edge ideals. The index of squarefree powers shows a quite different behavior than that of ordinary powers. Let $I$ be the edge ideal of a finite graph $G$. We denote the $k$th squarefree power of $I$ by $I^{[k]}$. It is clear that the unique minimal monomial set of generators of $I^{[k]}$ corresponds to the matchings of $G$ of size $k$. In particular, if $\nu(G)$ denotes the matching number of $G$, that is maximal size of a matching of $G$, then $\nu(G)$ coincides with the maximal number $k$ such that $I^{[k]}\neq 0$. In Theorem \[lin.quo\] we show that $I^{[\nu(G)]}$ always has linear quotients. In particular $\index(I^{[\nu(G)]})=\infty$ no matter whether or not $\index(I)=1$. Matchings with the property that one edge of the matching forms a gap with any other edge of the matching will be called a restricted matching. We denote by $\nu_0(G)$ the maximal size of a restricted matching of $G$. If there is no restricted matching we set $\nu_0(G)=1$. There are examples which show that $\nu(G)-\nu_0(G)$ may be arbitrary large. However for trees one can see that $\nu_0(G)\geq \nu(G)-1$. It is shown in Lemma \[levico\] that $\index(I^{[k]})=1$ for $k<\nu_0(G)$, and we conjecture that $\index(I^{[k]})>1$ for all $k\geq 