Conditions for the Yoneda algebra of a local ring to be generated in low degrees

The powers ${\mathfrak m}^n$ of the maximal ideal $\mathfrak m$ of a local Noetherian ring $R$ are known to satisfy certain homological properties for large values of $n$. For example, the homomorphism $R\to R/{\mathfrak m}^n$ is Golod for $n\gg 0$. We study when such properties hold for small values of $n$, and we make connections with the structure of the Yoneda Ext algebra, and more precisely with the property that the Yoneda algebra of $R$ is generated in degrees $1$ and $2$. A complete treatment of these properties is pursued in the case of compressed Gorenstein local rings.