AI 中文总结
该研究证明对所有 $b\ge1$，有序图 $Q_{2,b}$ 的相对Turán密度为 $1/2$，通过奇偶性分割构造实现下界，利用二元富级归约等方法得到上界，并给出路径 blow-up 值构建有序相对密度的可复用演算。
AI 中文摘要
对于每个 $b\ge1$，令 $Q_{2,b}$ 为从一个传递有序三角形的最右侧顶点附加一条长度为 $b$ 的单调尾得到的有序图。我们证明对所有 $b\ge1$，有 $\rho_{<}(Q_{2,b})=\frac{1}{2}$，因此此前孤立的情况 $Q_{2,2}$ 是一个精确的无限三角形尾族的成员。下界由精确的前向模板恒等式 $\lambda(Q_{2,b})=1/2$ 实现，且已在二元级宿主内部实现：奇偶性分割构造给出无 $Q_{2,b}$ 的二元级图，其在每个层级上的密度恰好为 $1/2$。上界使用二元富级归约，其关键输入是无 $Q_{2,b}$ 图的固有分解：分为尾起始顶点集 $T_b$ 和补集 $R$，满足从 $R$ 到 $T_b$ 无向前边、$G[T_b]$ 无有序三角形、$G[R]$ 无单调 $\vec{P}_{b+1}$。我们通过加权二元 $\vec{P}_{b+1}$ 平滑和加权二元 Mantel 平滑控制这些部分，其中后者源于二元超度量分割支配定理。我们还记录了精确的路径 blow-up 值，为有序相对密度提供了可复用的模板/富宿主演算。
英文摘要
For every $b\ge1$, let $Q_{2,b}$ be the ordered graph obtained from a transitive ordered triangle by attaching a monotone tail of length $b$ at its rightmost vertex. We prove $ρ_{<}(Q_{2,b})=\frac12$ for all $b\ge1$. Thus the previously isolated case $Q_{2,2}$ is one member of an exact infinite triangle-tail family. The lower bound is the sharp forward-template identity $λ(Q_{2,b})=1/2$, and it is realized already inside binary-level hosts: a parity-cut construction gives $Q_{2,b}$-free binary-level graphs with density exactly $1/2$ on every level. The upper bound uses the binary rich-level reduction. Its key input is the intrinsic decomposition of a $Q_{2,b}$-free graph into tail-starting vertices $T_b$ and the complement $R$: there are no forward edges from $R$ to $T_b$, $G[T_b]$ is ordered-triangle-free, and $G[R]$ is monotone-$\vec{P}_{b+1}$-free. We control these pieces by weighted binary $\vec{P}_{b+1}$ smoothing and weighted binary Mantel smoothing, the latter following from a binary ultrametric cut-domination theorem. We also record exact path-blow-up values, giving a reusable template/rich-host calculus for ordered relative densities.
