On applications of Razborov's flag algebra calculus to extremal 3-graph theory

Abstract

In this paper, we prove several new Tur\'an density results for 3-graphs with independent neighbourhoods. We show: \pi(K_4^-, C_5, F_{3,2})=12/49, \pi(K_4^-, F_{3,2})=5/18, and \pi(J_4, F_{3,2})=\pi(J_5, F_{3,2})=3/8, where J_t is the 3-graph consisting of a single vertex x together with a disjoint set A of size t and all $\binom{|A|}{2}$ 3-edges containing x. We also prove two Tur\'an density results where we forbid certain induced subgraphs: \pi(F_{3,2}, induced K_4^-)=3/8 and \pi(K_5, 5-set spanning 8 edges)=3/4. The latter result is an analogue for K_5 of Razborov's result that \pi(K_4, 4-set spanning 1 edge)=5/9. We give several new constructions, conjectures and bounds for Tur\'an densities of 3-graphs which should be of interest to researchers in the area. Our main tool is `Flagmatic', an implementation of Razborov's flag algebra calculus, which we are making publicly available. In a bid to make the power of Razborov's method more widely accessible, we have tried to make Flagmatic as user-friendly as possible, hoping to remove thereby the major hurdle that needs to be cleared before using the flag algebra calculus. Finally, we spend some time reflecting on the limitations of our approach, and in particular on which problems we may be unable to solve. Our discussion of the `complexity barrier' for the flag algebra calculus may be of general interest.