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.

The codegree threshold of K4−$K_4^{-}$

“…Our proof of Theorem 1.3 uses the flag algebra framework introduced by Razborov in [26], and in particular the semi-definite method first deployed by Razborov in [27]; see also [1,11,12,14] for expositions of the basic ideas. Such an approach is by now well established in extremal hypergraph theory, and since a treatment of the general theory of flag algebras is outside the scope of this article, we content ourselves here with giving brisk definitions of some of the standard terms and concepts of flag algebras that we shall use, and refer an interested reader to the papers cited above for further details and discussion.…”