A container theorem for general digraphs with forbidden subdigraphs

TL;DR

This paper develops a new container theorem for general digraphs with certain sparsity conditions, enabling asymptotic enumeration and structural analysis of H-free digraphs, extending previous results.

Contribution

It introduces a container theorem applicable to all digraphs under a natural sparsity condition, broadening the scope of hypergraph container methods.

Findings

Provides asymptotic counting for H-free digraphs.

Describes the typical structure of H-free digraphs.

Unifies and extends previous results in digraph enumeration.

Abstract

In a seminal work, K\"uhn, 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. For the edge-weight parameter $a=2$, this condition permits digraphs with $2$-cycles (density at most $1$) but excludes denser obstructions like $DK_3$; for larger $a$ it 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.