On oriented Tur\'an problems

TL;DR

This paper investigates the maximum number of arcs in large oriented graphs avoiding certain subgraphs, providing exact values, stability results, and extending classical theorems to the oriented graph setting.

Contribution

It determines exact Turán numbers for all small oriented graphs with up to three arcs and large n, and extends key theorems to oriented graphs.

Findings

Exact Turán numbers for all oriented graphs with up to three arcs.

A stability result for oriented graphs avoiding specific subgraphs.

An oriented version of the F"{u}redi-Alon-Krivelevich-Sudakov Theorem.

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.