On the Local Structure of Oriented Graphs - a Case Study in Flag Algebras

Abstract

Let $G$ be an $n$-vertex oriented graph. Let $t(G)$ (respectively $i(G)$) be the probability that a random set of $3$ vertices of $G$ spans a transitive triangle (respectively an independent set). We prove that $t(G) + i(G) \geq \frac{1}{9}-o_n(1)$. Our proof uses the method of flag algebras that we supplement with several steps that make it more easily comprehensible. We also prove a stability result and an exact result. Namely, we describe an extremal construction, prove that it is essentially unique, and prove that if $H$ is sufficiently far from that construction, then $t(H) + i(H)$ is significantly larger than $\frac{1}{9}$.

We go to greater technical detail than is usually done in papers that rely on flag algebras. Our hope is that as a result this text can serve others as a useful introduction to this powerful and beautiful method.