Applications of the Semi-Definite Method to the Turán Density Problem for 3-Graphs

Abstract

In this paper, we prove several new Turán density results for 3-graphs with independent neighbourhoods. We show:\begin{align*} \pi (K_4^-, C_5, F_{3,2})=12/49, \pi (K_4^-, F_{3,2})=5/18 \textrm {and} \pi (J_4, F_{3,2})=\pi (J_5, F_{3,2})=3/8, \end{align*}whereJtis the 3-graph consisting of a single vertexxtogether with a disjoint setAof sizetand all$\binom{|A|}{2}$3-edges containingx. We also prove two Turán density results where we forbid certain induced subgraphs:\begin{align*} \pi (F_{3,2}, \textrm { induced }K_4^-)=3/8 \textrm {and} \pi (K_5, 5\textrm {-set spanning exactly 8 edges})=3/4. \end{align*}The latter result is an analogue forK5of Razborov's result that\begin{align*} \pi (K_4, 4\textrm {-set spanning exactly 1 edge})=5/9. \end{align*}We give several new constructions, conjectures and bounds for Turán densities of 3-graphs which should be of interest to researchers in the area. Our main tool is ‘Flagmatic’, an implementation of Razborov's semi-definite method, 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 semi-definite method. 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 semi-definite method may be of general interest.

Flag Algebras: An Interim Report

“…Falgas-Ravry and Vaughan [FRV13] proved, besides the results we already cited above in various contexts, several more exact results: …”

“…Then FalgasRavry and Vaughan [FRV13], also using flag algebras, proved that the pairs (G 3 , F 3,2 ) and (K 3 4 , J 4 ) are non-principal. The former example is remarkable since they were also able to compute…”

“…[Raz10] that was verified in [BT11] and later in [FRV13] using the flagmatic software (we will discuss the latter in Section 4.1). The scale of this improvement reflects a general phenomenon: let me cautiously suggest that I…”

Quasi‐carousel tournaments

“…Using the flag algebra semidefinite method, we were able to obtain the bound subject to floating point rounding errors. This suggests that the conjecture is true and that there may be a straightforward (but numerically intensive) proof using the semidefinite method and rounding techniques (see for some examples).…”

“…Using the flag algebra semidefinite method, we were able to obtain the bound ∀ ∈ Hom + ( 0 , ℝ), ( 4 ) ⩽ 0.157516, subject to floating point rounding errors. This suggests that the conjecture is true and that there may be a straightforward (but numerically intensive) proof using the semidefinite method and rounding techniques (see [1,10,11,14,22] for some examples). The intuition of the recursive construction of , is that at every step we have one part −1 ⧵ that maximizes the density of ⃗ 3 (hence is almost balanced) and another part whose vertices all beat the first part.…”