AI 中文总结
本文证明任意具有零ℓ-度Turán密度的k-均匀超图是分层的，k=3时证实了Ding等人关于3-均匀超图代码度Turán密度的猜想。
AI 中文摘要
代码度Turán密度π_co(F)是指满足以下条件的所有γ∈[0,1)的上确界：对于任意足够大的n，存在n个顶点的不含F的k-均匀超图H，其每个(k-1)顶点子集都包含在至少γn条边中。Ding、Lamaison、Liu、Wang和Yang（JLMS，2025）研究了哪些3-均匀超图F满足π_co(F)=0的问题，他们引入了分层3-均匀超图的概念，并猜想一个3-均匀超图具有零代码度Turán密度当且仅当它是分层的且具有零均匀Turán密度。对于k≥3，一个k-均匀超图被称为分层的，当它的顶点可以被标记，使得每条边都有唯一的最大标记，且具有相同最大标记的两条边具有相同的标记多重集。本文证明，任意m个顶点的非分层k-均匀超图F满足π_co(F)≥q_{k,m}^{-q_{k,m}}>0，其中q_{k,m}=(k-1)^{m+1}-1)/(k-2)，这意味着任意具有零ℓ-度Turán密度的k-均匀超图都是分层的，且k=3的情况证实了Ding等人的猜想。
英文摘要
The codegree Turán density $π_{\mathrm{co}}(F)$ is the supremum over all $γ\in [0,1)$ such that, for arbitrarily large $n$, there exists an $n$-vertex $F$-free $k$-graph $H$ whose every $(k-1)$-subset of vertices lies in at least $γn$ edges. Ding, Lamaison, Liu, Wang, and Yang (JLMS, 2025) studied the problem of what 3-graphs $F$ satisfy $π_{\mathrm{co}}(F) = 0$. They introduced layered $3$-graphs and conjectured that a $3$-graph has zero codegree Turán density if and only if it is layered and has zero uniform Turán density. For $k\ge 3$, a $k$-graph is called layered if its vertices can be labelled so that every edge has a unique maximum label and two edges with the same maximum label have the same label multiset. In this paper, we show that every non-layered $k$-graph $F$ on $m$ vertices satisfies \[ π_{\mathrm{co}}(F)\ge q_{k,m}^{-q_{k,m}}>0, \quad \text{where}\quad q_{k,m}=\frac{(k-1)^{m+1}-1}{k-2}, \] which implies any $k$-graph with zero $\ell$-degree Turán density is layered, and the case $k=3$ confirms the conjecture of Ding, Lamaison, Liu, Wang, and Yang.
