# Typical structure of oriented graphs and digraphs with forbidden blow-up transitive triangle

## TL;DR

This paper extends the Erdős-Stone theorem to weighted digraphs, demonstrating that almost all oriented graphs and digraphs avoiding a blow-up transitive triangle are nearly bipartite, using the Regularity Lemma.

## Contribution

It provides a stability result for oriented graphs and digraphs with forbidden blow-up transitive triangles, establishing their near bipartiteness.

## Key findings

- Almost all such graphs are nearly bipartite.
- The Regularity Lemma is used to prove the stability result.
- An analogue of the Erdős-Stone theorem is established for weighted digraphs.

## Abstract

In this work, we establish an analogue result of the Erd\"os-Stone theorem of weighted digraphs using Regularity Lemma of digraphs. We give a stability result of oriented graphs and digraphs with forbidden blow-up transitive triangle and show that almost all oriented graphs and almost all digraphs with forbidden blow-up transitive triangle are almost bipartite respectively.