Stein’s minimum-semidegree conjecture for oriented paths

Prove that for every positive integer k, every oriented graph D with minimum semidegree greater than k/2 contains every oriented path of length k, thereby resolving Stein’s conjecture for all oriented paths.

Background

Stein’s conjecture asserts that a minimum semidegree exceeding half the target path length should force every orientation of a path of that length in an oriented graph. The threshold is best possible because a disjoint union of regular tournaments on k vertices has minimum semidegree at most k/2 and cannot contain a path of length k.

The paper notes that the conjecture is known for several subclasses, including directed paths, anti-directed paths, and—according to the cited recent work—oriented paths with two blocks. Nevertheless, the conjecture remains unresolved in its full generality, including orientations with more complex block structures.