On oriented Tur\'an problems

Abstract

The oriented Tur\'{a}n number of a given oriented graph $\overrightarrow{F}$, denoted by $\exo(n,\overrightarrow{F})$, is the largest number of arcs in $n$-vertex $\overrightarrow{F}$-free oriented graphs. This concept could be seen as an oriented version of the classical Tur\'{a}n number. In this paper, we first prove several propositions that give exact results for several oriented graphs. In particular, we determine all exact values of $\exo(n,\overrightarrow{F})$ for every oriented graph $\overrightarrow{F}$ with at most three arcs and sufficiently large $n$. After that, we prove a stability result and use it to determine the Tur\'an number of an orientation of $C_4$. Finally, we prove oriented versions of the random zooming theorem by Fern\'andez, Hyde, Liu, Pikhurko and Wu and the almost regular subgraph theorem by Erd\H{o}s and Simonovits, and use them to obtain an oriented version of the F\"{u}redi-Alon-Krivelevich-Sudakov Theorem, which generalizes the famous KST Theorem.