AI 中文总结
研究r-一致超图族的(r-2)-一致Turán密度集合，证明足够大的r下密度在4r^{-r}处发生相变，低于阈值为可数代数集，高于为全区间，且集合非闭。
AI 中文摘要
受普通Turán问题和ℓ度Turán问题平行发展的启发，我们研究了可能无限的r-一致超图族所对应的(r-2)-一致Turán密度集合。我们证明，对于所有足够大的r，这些密度在4r^{-r}处表现出相变：低于该阈值的密度构成一个可数的代数值集合，而高于该阈值则覆盖整个区间[4r^{-r},1]。对于每个r≥3，低于阈值的密度恰好是该范围内的有限调色板Lagrangian值，并且每个密度都由一个有限的禁止族实现。通过改编Pikhurko的方法，我们证明对于每个r≥3，一致密度集合在阈值以下遗漏了其极限点的连续多个值，因此该集合不是闭集。
英文摘要
Motivated by parallel developments in ordinary and $\ell$-degree Turán problems, we study the set of $(r-2)$-uniform Turán densities of possibly infinite families of $r$-uniform hypergraphs. We prove that, for all sufficiently large $r$, these densities exhibit a phase transition at $4r^{-r}$, from a countable set of algebraic values below this threshold to the full interval $[4r^{-r},1]$. For every $r\ge3$, the densities below the threshold are exactly the finite-palette Lagrangians in that range, and each is realized by a finite forbidden family. Adapting a method of Pikhurko, we show that, for every $r\ge3$, the set of uniform densities omits continuum many of its limit points below the threshold and is therefore not closed.
