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PreprintCC BY 4.0arXiv

A New Upper Bound for the Turán Density of the Tetrahedron

  • Gyeongwon Jeong
  • Seonghun Park
  • Seonghyuk Im
  • Joonkyung Lee
  • Hongseok Yang
Open at sourcePDFDOICite

Abstract

We prove that the Turán density of the tetrahedron K4(3) satisfies π(K4(3)) ≤ 14993367693127837/26880000000000000 < 0.557789, 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.

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Cite

BibTeXRIS
BibTeX
@misc{jeong2026upper,
  title = {{A New Upper Bound for the Turán Density of the Tetrahedron}},
  author = {Gyeongwon Jeong and Seonghun Park and Seonghyuk Im and Joonkyung Lee and Hongseok Yang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.27495},
  eprint = {2609.27495},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.27495},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A New Upper Bound for the Turán Density of the Tetrahedron
AU  - Gyeongwon Jeong
AU  - Seonghun Park
AU  - Seonghyuk Im
AU  - Joonkyung Lee
AU  - Hongseok Yang
PY  - 2026
DA  - 2026/10/02
PB  - arXiv
DO  - 10.48550/arxiv.2609.27495
UR  - https://arxiv.org/abs/2609.27495
AB  - We prove that the Turán density of the tetrahedron K_4^(3) satisfies π(K_4^(3)) ≤ 14993367693127837/26880000000000000 < 0.557789, 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.
ER  -