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On the supersaturation of oriented Turán problems

Xuanrui Hu, Yuefang Sun

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 parameter could be seen as a natural oriented version of the classical Turán number. In this paper, we study the supersaturation phenomenon for oriented Turán problems, and prove oriented versions of the famous Erdős-Simonovits Supersaturation Theorem and Moon-Moser inequality, and supersaturation theorems for transitive tournaments and antidirected complete bipartite graphs.

On the supersaturation of oriented Turán problems

Abstract

The oriented Turán number of a given oriented graph , denoted by , is the largest number of arcs in -vertex -free oriented graphs. This parameter could be seen as a natural oriented version of the classical Turán number. In this paper, we study the supersaturation phenomenon for oriented Turán problems, and prove oriented versions of the famous Erdős-Simonovits Supersaturation Theorem and Moon-Moser inequality, and supersaturation theorems for transitive tournaments and antidirected complete bipartite graphs.
Paper Structure (6 sections, 10 theorems, 61 equations)

This paper contains 6 sections, 10 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Our results
  3. Proof of Theorem \ref{['superori']}
  4. Proof of Theorem \ref{['TT3']}
  5. Proofs of Theorems \ref{['OMM']} and \ref{['TTdensity']}
  6. Proof of Theorem \ref{['antidirected4-cycle']}

Key Result

Theorem 1.4

For any acyclic oriented graph $\overrightarrow{F}$, we have

Theorems & Definitions (28)

  • Definition 1.1: Oriented Turán number
  • Definition 1.2: Oriented Turán density
  • Definition 1.3: Oriented edge density
  • Theorem 1.4: Valadkhan vala
  • Theorem 1.5: Füredi-Alon-Krivelevich-Sudakov FurediAlon-Krivelevich-Sudakov
  • Theorem 1.6: Gerbner-Hu-Sun Gerbner-Hu-Sun
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 18 more