First Nonlinear Syzygies of Ideals Associated to Graphs
  Consider an ideal $I\subset K[x_1,..., x_n]$, with $K$ an arbitrary field, generated by monomials of degree two. Assuming that $I$ does not have a linear resolution, we determine the step $s$ of the minimal graded free resolution of $I$ where nonlinear syzygies first appear, we show that at this step of the resolution nonlinear syzygies are concentrated in degree $s+3$, and we compute the corresponding graded Betti number $\beta_{s,s+3}$. The multidegrees of these nonlinear syzygies are also determined and the corresponding multigraded Betti numbers are shown to be all equal to 1. 
Oscar Fernandez-Ramos, Philippe Gimenez