Prime-regular subgraph from Boolean polynomial constraints

For prime $p$, a graph with $m>(p-1)n$ edges and maximum degree at most $2p-1$ contains a nonempty $p$-regular subgraph. Over ${\mathbb{F}}_p$, apply the Combinatorial Nullstellensatz on the Boolean edge-variable cube toIts full squarefree coefficient is nonzero because the first product has degree below $m$. A nonzero value cannot occur at the zero vector; at any other Boolean vector the second product vanishes, and the first forces all selected degrees to be multiples of $p$. The maximum-degree condition makes every positive selected degree exactly $p$.