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We use relations in the tautological ring of the moduli spaces$\overline{\mathcal{M}}_{g,n}$ derived by Pandharipande, Pixton, and Zvonkinefrom the Givental formula for the $r$-spin Witten class in order to obtain somerestrictions on the dimensions of the tautological rings of the open modulispaces $\mathcal{M}_{g,n}$. In particular, we give a new proof for the resultof Looijenga (for $n=1$) and Buryak et al. (for $n\geq 2$) that $\dimR^{g-1}(\mathcal{M}_{g,n}) \leq n$. We also give a new proof of the result ofLooijenga (for $n=1$) and Ionel (for arbitrary $n\geq 1$) that$R^{i}(\mathcal{M}_{g,n}) =0$ for $i\geq g$ and give some estimates for thedimension of $R^{i}(\mathcal{M}_{g,n})$ for $i\leq g-2$.

The tautological ring of $\mathcal{M}_{g,n}$ via Pandharipande-Pixton-Zvonkine $r$-spin relations