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A container theorem for general digraphs with forbidden subdigraphs

Jianxi Liu

Abstract

In a seminal work, Kühn, Osthus, Townsend, and Zhao used the hypergraph container method to determine the typical structure of oriented graphs and digraphs avoiding a fixed tournament or cycle. Their main tool, a container theorem for oriented graphs, does not directly extend to all digraphs due to the existence of counterexamples such as the double triangle $DK_3$. In this paper we prove a container theorem for general digraphs under a natural sparsity condition that, for the edge-weight parameter $a=2$, reduces to the oriented case, but for larger $a$ allows digraphs with a controlled density of 2-cycles. As applications, we obtain asymptotic counting results for $H$-free digraphs and describe the typical structure of digraphs avoiding a fixed digraph $H$ satisfying our condition. Our results unify and extend several previous results in the area.

A container theorem for general digraphs with forbidden subdigraphs

Abstract

In a seminal work, Kühn, Osthus, Townsend, and Zhao used the hypergraph container method to determine the typical structure of oriented graphs and digraphs avoiding a fixed tournament or cycle. Their main tool, a container theorem for oriented graphs, does not directly extend to all digraphs due to the existence of counterexamples such as the double triangle . In this paper we prove a container theorem for general digraphs under a natural sparsity condition that, for the edge-weight parameter , reduces to the oriented case, but for larger allows digraphs with a controlled density of 2-cycles. As applications, we obtain asymptotic counting results for -free digraphs and describe the typical structure of digraphs avoiding a fixed digraph satisfying our condition. Our results unify and extend several previous results in the area.
Paper Structure (10 sections, 6 theorems, 20 equations)

This paper contains 10 sections, 6 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Digraphs and weighted edges
  4. The hypergraph container theorem
  5. The auxiliary hypergraph
  6. Key lemmas
  7. Degree estimate
  8. Supersaturation
  9. Proof of Theorem \ref{['thm:main']}
  10. Discussion: necessity of the condition and open problems

Key Result

Theorem 1.1

Let $H$ be an oriented graph with $h=v(H)$ and $e(H)\ge 2$, and let $a\ge 1$. For every $\varepsilon>0$ there exists $c>0$ such that for all sufficiently large $N$ there exists a collection $\mathcal{C}$ of digraphs on $[N]$ with the following properties.

Theorems & Definitions (8)

  • Theorem 1.1: KOTZ, Theorem 3.3
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 2.1: Saxton–Thomason SAX2015, Corollary 2.7
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof