Two-block paths in oriented graphs of large semidegree

We study the existence of oriented paths with two blocks in oriented graphs under semidegree conditions. A block of an oriented path is a maximal directed subpath. Given positive integers $k$ and $\mathrm{\ell }$ with $k\mathrm{/}2\le \mathrm{\ell }<k$, we establish a semidegree function that guarantees the containment of every oriented path with two blocks of sizes $\mathrm{\ell }$ and $k-\mathrm{\ell }$. As a corollary, we show that every oriented graph with all in- and out-degrees at least $3k\mathrm{/}4$ contains every two-block path with $k$ arcs. Our results extend previous work on Stein’s conjecture and related problems concerning oriented paths.