On subgraphs with degrees of prescribed residues in the random graph

We show that with high probability the random graph Gn,1/2$$ {G}_{n,1/2} $$ has an induced subgraph of linear size, all of whose degrees are congruent to r(modq)$$ r\kern0.3em \left(\operatorname{mod}\kern0.3em q\right) $$ for any fixed r$$ r $$ and q≥2$$ q\ge 2 $$ . More generally, the same is true for any fixed distribution of degrees modulo q$$ q $$ . Finally, we show that with high probability we can partition the vertices of Gn,1/2$$ {G}_{n,1/2} $$ into q+1$$ q+1 $$ parts of nearly equal size, each of which induces a subgraph all of whose degrees are congruent to r(modq)$$ r\kern0.3em \left(\operatorname{mod}\kern0.3em q\right) $$ . Our results resolve affirmatively a conjecture of Scott, who addressed the case q=2$$ q=2 $$ .