Extend the minimum-degree bound to odd target degrees

Extend Theorem 1.1 for even target degrees to odd values of the target minimum degree s, thereby obtaining an analogous bound for small subgraphs of large minimum degree when s is odd.

Background

The paper proves the conjectured polylogarithmic upper bound for the order of a small subgraph of minimum degree at least s when s is even. It separately resolves the bounded-order problem for odd s≥5, but does not establish the corresponding general quantitative bound for odd target degrees. The authors identify extending the even-degree theorem to odd s as the principal remaining problem.