AI 中文总结
该研究解决了埃尔德什1977年提出的四边相交问题，证明足够大的n顶点图的f(n,4)=2n-4，基于核心-缓冲区框架结合新增的高度核心与刚性分析完成证明。
AI 中文摘要
对于一个n顶点图G及其顶点集的一个排列σ，令σ(G)表示G的对应重标记，记I_G(σ)=|E(G)∩E(σ(G))|。设f(n,k)为满足对每个σ都有I_G(σ)≥k的n顶点图的最小边数。1977年，埃尔德什询问是否有f(n,4)=2n-4，并指出K_{2,n-2}给出了上界。我们证明，对所有足够大的n，f(n,4)=2n-4，等价于每个边数最多为2n-5的足够大的n顶点图，都存在一个重标记使得公共边数最多为3。我们的证明受Fang和Hou关于埃尔德什-马伦五边相交问题的近期工作启发，基于他们的核心-缓冲区和吸收框架。主要新增内容是一个增长的高度核心C，满足|C|Δ(G-C)=o(n)，以及对相关一阶矩估计中等式情况的刚性分析。该分析表明，唯一强制出现四个局部公共边的核心-缓冲区构型是K_{2,|C|}类型；严格边界e(G)≤2n-5则提供了一个缺陷，可打破该构型。
英文摘要
For an $n$-vertex graph $G$ and a permutation $σ$ of its vertex set, let $σ(G)$ denote the corresponding relabelling of $G$, and put \[ I_G(σ)=|E(G)\cap E(σ(G))|. \] Let $f(n,k)$ be the minimum number of edges in an $n$-vertex graph for which $I_G(σ)\geq k$ for every $σ$. In 1977 Erdős asked whether $f(n,4)=2n-4$, observing that $K_{2,n-2}$ gives the upper bound. We prove that, for all sufficiently large $n$, \[ f(n,4)=2n-4. \] Equivalently, every sufficiently large $n$-vertex graph with at most $2n-5$ edges has a relabelling with at most three common edges. Our proof is inspired by the recent work of Fang and Hou on the Erdős--Mullin five-edge intersection problem and builds on their core--buffer and absorption framework. The main additional ingredients are a growing high-degree core $C$ satisfying \[ |C|Δ(G-C)=o(n), \] and a rigidity analysis of the equality case in the relevant first-moment estimate. This analysis shows that the only core--buffer configuration forcing four local common edges is of $K_{2,|C|}$ type; the strict bound $e(G)\leq2n-5$ then supplies a defect which breaks this configuration.
