Find bounded-order regular subgraphs of prescribed odd degree

Establish that sufficiently dense graphs contain a bounded-order s-regular subgraph, strengthening the bounded-order result that currently guarantees only a subgraph of minimum degree at least s.

Background

The paper's bounded-order theorem for odd s≥5 guarantees a subgraph on boundedly many vertices with minimum degree at least s, while the analogous bounded-order existence of an s-regular subgraph is stronger. The authors note that this regular-subgraph problem is known in some cases, including even s and s=3, but remains unresolved in the general odd-degree setting addressed by the paper.