Ramsey--Tur\'an Factors of Non-directed Oriented Cycles in Oriented Graphs

Let $\overrightarrow{C}$ be any orientation of the cycle $C_{\mathrm{\ell }}$ which is not directed. We prove that, for every integer $\mathrm{\ell }\ge 3$ and every $\mu >0$, there is a real $\gamma$ such that every sufficiently large oriented graph $D$ with $\mathrm{\ell }\mid \mathrm{\mid }D\mathrm{\mid }$, minimum semidegree at least $(1\mathrm{/}4+\mu )\mathrm{\mid }D\mathrm{\mid }$ and independence number at most $\gamma \mathrm{\mid }D\mathrm{\mid }$ has a $\overrightarrow{C}$-factor. The constant $1\mathrm{/}4$ is asymptotically tight. This proof establishes Ramsey-Tur'an type lattice absorption lemmas and an almost covering theorem via the oriented tree embedding lemma under chromatic number constraints.