On oriented Turán problems
Dániel Gerbner, Xuanrui Hu, Yuefang Sun
TL;DR
The paper studies oriented Turán problems by developing a compressibility-based framework and proving exact and asymptotic results for $ex_o(n, \overrightarrow{F})$. The authors derive exact values for all $\overrightarrow{F}$ with at most three arcs for large $n$, establish stability near Turán graphs, determine the Turán number for an orientation of $C_4$, and prove oriented analogues of the random zooming and almost-regular subgraph theorems, yielding a generalized oriented Füredi–Alon–Krivelevich–Sudakov type bound. The work connects the oriented problem to underlying undirected Turán theory via the abstract chromatic number and provides new tools for embedding and regularization in bipartite orientations. These results advance understanding of how orientation constraints shape extremal arc counts and offer techniques potentially applicable to broader orientation-constraint problems.
Abstract
The oriented Turán 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 Turán 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 Turán number of an orientation of $C_4$. Finally, we prove oriented versions of the random zooming theorem by Fernández, Hyde, Liu, Pikhurko and Wu and the almost regular subgraph theorem by Erdős and Simonovits, and use them to obtain an oriented version of the Füredi-Alon-Krivelevich-Sudakov Theorem, which generalizes the famous KST Theorem.