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

Let G be an n-vertex oriented graph. Let t(G) (respectively i(G)) be the prob-ability that a random set of 3 vertices of G spans a transitive triangle (respectively an independent set). We prove that t(G) + i(G) >= 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 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.