AI 中文总结
针对埃尔德什与马伦提出的五边相交问题，通过结合近通用顶点定量排除等方法，证明了足够大n时满足μ(G)≥5的n顶点图最少边数为2n−2，解答了该问题。
AI 中文摘要
对于n顶点图G及其顶点集的一个排列π，令I_G(π)=|E(G)∩E(πG)|，μ(G)=min_π I_G(π)。设f(n,k)为满足μ(G)≥k的n顶点图G的最少边数。埃尔德什记录了马伦的一个构造，表明f(n,5)≤2n−2，并询问对于足够大的n，等式是否成立。我们证明该等式成立：对于所有足够大的n，f(n,5)=2n−2。等价地，每个边数最多为2n−3的足够大的n顶点图，都存在一个重排，使得公共边数不超过4。该证明结合了近通用顶点的定量排除、具有低度缓冲顶点的有限高度核心、列表 packing 以及Lovász局部引理的稀疏排列版本。
英文摘要
For an $n$-vertex graph $G$ and a permutation $π$ of its vertex set, let \[ I_G(π)=|E(G)\cap E(G_π)|,\qquad μ(G)=\min_π I_G(π), \] where $G_π$ is the copy of $G$ obtained by relabelling every vertex $x\in V(G)$ as $π(x)$. Let $f(n,k)$ be the minimum number of edges in an $n$-vertex graph $G$ satisfying $μ(G)\ge k$. Erdős recorded a construction of Mullin showing $f(n,5)\le 2n-2$ and asked whether equality holds for sufficiently large $n$. We prove that it does: \[ f(n,5)=2n-2 \] for all sufficiently large $n$. The proof strategy is a core--buffer--completion framework: it moves the few high-degree vertices into carefully chosen low-degree positions, confines the allowed overlap to this bounded part, and then relabels the sparse remainder without creating any additional common edge.
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