On the supersaturation of oriented Tur\'an problems

TL;DR

This paper explores the supersaturation phenomenon in oriented Turán problems, establishing new supersaturation theorems for various oriented graphs, extending classical results to the oriented setting.

Contribution

It introduces oriented versions of the Erdős-Simonovits Supersaturation Theorem and Moon-Moser inequality, advancing the understanding of extremal properties in oriented graphs.

Findings

Proved oriented Erdős-Simonovits Supersaturation Theorem.

Established supersaturation results for transitive tournaments.

Extended classical inequalities to the oriented graph context.

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 parameter could be seen as a natural oriented version of the classical Tur\'{a}n number. In this paper, we study the supersaturation phenomenon for oriented Tur\'{a}n problems, and prove oriented versions of the famous Erd\H{o}s-Simonovits Supersaturation Theorem and Moon-Moser inequality, and supersaturation theorems for transitive tournaments and antidirected complete bipartite graphs.