发表机构
Leipzig University; Max Planck Institute for Mathematics in the Sciences; Universidad Pública de Navarra; Institute for Basic Science; Universitat Politècnica de Catalunya (UPC)（莱比锡大学; 马普科学数学研究所; 纳瓦拉公立大学; 基础科学研究院; 加泰罗尼亚理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了均匀图兰密度中四面体超图K4^(3)的密度为1/2，解决了图兰四面体问题在均匀约束下的情形，并证实了勒德尔1986年下界构造的最优性。
AI 中文摘要
图兰四面体问题要求确定完全超图$K_4^{(3)}$（四面体）的图兰密度。该问题由图兰于1941年提出，是极值组合学中最著名的问题之一，其解决将获得埃尔德什提供的500美元奖金。在20世纪80年代，埃尔德什和绍什提出，当边被约束为在宿主超图中均匀分布时，确定$K_4^{(3)-}$（破四面体）和$K_4^{(3)}$（四面体）的图兰密度。破四面体的情况据推测较容易，已由格列博夫、克拉尔和沃莱茨 [以色列数学杂志 211 (2016), 349-366] 以及赖赫尔、勒德尔和沙赫特 [欧洲数学学会杂志 20 (2018), 1139-1159] 解决。我们通过证明$K_4^{(3)}$的均匀图兰密度等于1/2来解决四面体的情况；这证实了勒德尔1986年的下界构造是最优的。
英文摘要
Turán's Tetrahedron Problem asks to determine the Turán density of the complete hypergraph $K_4^{(3)}$ (tetrahedron). This problem, posed by Turán in 1941, is one of the most famous problems in extremal combinatorics and its solution would attract \$500 prize from Erdős. In the 1980s, Erdős and Sós asked to determine Turán densities of $K_4^{(3)-}$ (broken tetrahedron) and $K_4^{(3)}$ (tetrahedron) when edges are constrained to be uniformly distributed in the host hypergraph. The presumably easier case of the broken tetrahedron was solved by Glebov, Král' and Volec [Israel J. Math. 211 (2016), 349-366] and Reiher, Rödl and Schacht [J. Eur. Math. Soc. 20 (2018), 1139-1159]. We solve the tetrahedron case by proving that the uniform Turán density of $K_4^{(3)}$ is equal to 1/2; this confirms that Rödl's lower bound construction from 1986 is optimal.
CommentsSome of the proofs in this paper are computer-assisted; the code and data are provided as ancillary files
