Let $G$ be an $n$-vertex graph with $e(G)\ge n^2/k$. We prove a self-contained internal short-cycle core theorem at the threshold $k\le n^{1/3}$: the graph $G$ contains a subgraph $H_6$ with $Ω(n^2/k^3)$ edges in which every two distinct edges lie together on a cycle of length at most $6$ contained in $H_6$, and a subgraph $H_8$ with $Ω(n^2/k^2)$ edges in which every two distinct edges lie together on a cycle of length at most $8$ contained in $H_8$. In density notation $ρ=e(G)/n^2$, this gives internal cores of sizes $Ω(ρ^3n^2)$ and $Ω(ρ^2n^2)$ throughout the range $ρ\ge n^{-1/3}$. The $C_{\le6}$ conclusion above is an edge-connected statement and does not impose the adjacent-edge $C_4$ condition appearing in the strongest Duke--Erdős--Rödl formulation. We also include two complementary results clarifying this distinction. First, under the ambient-witness convention, every graph with at least $n^2/k$ edges and $k=o(n^{1/2})$ contains $Ω(n^2/k^3)$ selected edges whose pairs are witnessed by ambient cycles of length at most $6$, with adjacent pairs witnessed by ambient $C_4$'s. Second, under the standard internal strong $C_6$ convention, for every fixed $β\in[1/3,1/2)$ there is an infinite sequence of bipartite graphs $G$ with $n\to\infty$ and $e(G)=Θ_β(n^{2-β})$ such that every internally strongly $C_6$-connected subgraph has only $O_β(ρ(G)^3n^2/(\log n)^2)$ edges. The obstruction is a random cyclic shift-lift of $K_{q,q}$, together with an occupancy estimate excluding large aligned two-covers.

翻译：设 $G$ 为 $n$ 顶点图，满足 $e(G)\ge n^2/k$。我们在门槛 $k\le n^{1/3}$ 处证明了一个自包含的内部短圈核定理：图 $G$ 包含一个子图 $H_6$，具有 $\Omega(n^2/k^3)$ 条边，其中 $H_6$ 内任意两条不同的边共同位于一条长度不超过 $6$ 且包含于 $H_6$ 的圈上；同时存在一个子图 $H_8$，具有 $\Omega(n^2/k^2)$ 条边，其中 $H_8$ 内任意两条不同的边共同位于一条长度不超过 $8$ 且包含于 $H_8$ 的圈上。在密度符号 $\rho=e(G)/n^2$ 下，这在 $\rho\ge n^{-1/3}$ 范围内分别给出了大小为 $\Omega(\rho^3n^2)$ 和 $\Omega(\rho^2n^2)$ 的内部核。上述关于 $C_{\le6}$ 的结论是一个边连通性的陈述，并未施加最强的Duke–Erdős–Rödl形式中所出现的相邻边 $C_4$ 条件。我们还附加了两个互补结果以澄清这一区别。首先，在环境见证约定下，每个具有至少 $n^2/k$ 条边且 $k=o(n^{1/2})$ 的图包含 $\Omega(n^2/k^3)$ 条选定边，其边对由长度不超过 $6$ 的环境圈见证，且相邻边对由环境 $C_4$ 见证。其次，在标准内部强 $C_6$ 约定下，对于每个固定的 $\beta\in[1/3,1/2)$，存在一个无限序列的二部图 $G$，满足 $n\to\infty$ 且 $e(G)=\Theta_\beta(n^{2-\beta})$，使得每个内部强 $C_6$ 连通子图仅有 $O_\beta(\rho(G)^3n^2/(\log n)^2)$ 条边。此障碍源于 $K_{q,q}$ 的随机循环移位提升，并结合了排除大型对齐二覆盖的占用估计。