A New Upper Bound for the Turán Density of the Tetrahedron

2026-09-23

AI Summary

本文通过结合切割平面和列生成技术，使用精确的七顶点标志代数证书改进了四面体图的Turán密度上界至0.557808。

Abstract

We prove that the Turán density of the tetrahedron $K_4^{(3)}$ satisfies $π(K_4^{(3)}) \le 312372062889819/560000000000000 < 0.557808$, improving Baber's upper bound of $0.5615$ and closing about $62\%$ of the gap to the conjectured value $5/9$. The proof uses an exact seven-vertex flag-algebra certificate incorporating degree-stationarity from Razborov's differential method. To find the certificate, we combine the established techniques of cutting planes and column generation to optimize jointly over flag families whose types have at most five vertices. We give a complete formal proof of this Turán density bound in Lean 4.

Problem

• Turán density
• tetrahedron
• upper bound

Innovation

• Turán density
• flag-algebra certificate
• degree-stationarity
• cutting planes and column generation
• formal proof in Lean 4