埃尔德什四边相交问题的渐近解
An asymptotic solution to the Erdős four-edge intersection problem
AI总结：
本文渐近解决埃尔德什四边相交问题，证明f(n,4)在2n-10n^{2/3}-7与2n-4之间，即f(n,4)=2n-o(n)，在次线性误差内验证了埃尔德什的猜想。
AI中文摘要：
对于n个顶点的图G及其顶点集的一个排列σ，令σ(G)表示G经重标记后的对应图，记I_G(σ)=|E(G)∩E(σ(G))|。设f(n,k)为n顶点图中，对任意排列σ都满足I_G(σ)≥k的图的最小边数。1977年，埃尔德什在提出该问题时讨论了k的较小值，将k=4和k=5的情形留作后续自然的开放问题。对于k=4，他提出f(n,4)=2n-4，该上界由完全二部图K_{2,n-2}实现；相邻的k=5问题最近已由方和侯完全解决。本文证明，所有阶数为n、边数至多为2n-10n^{2/3}-7的图G，都存在一个排列使得公共边数至多为3。由此可得2n-10n^{2/3}-7<f(n,4)≤2n-4，进而推出f(n,4)=2n-o(n)。因此，本文渐近解决了埃尔德什四边相交问题，在次线性误差项内验证了埃尔德什提出的数值。作为对比，对于所有足够大的n，方和侯的结果保证，边数至多为2n-3的图的公共边数至多为4；而仅将边数界减少10n^{2/3}+4=o(n)，就可保证公共边数至多为3。
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 his 1977 formulation of the problem, Erdős discussed the small values of $k$ and left the cases $k=4$ and $k=5$ as the next natural open questions. For $k=4$ he asked whether $f(n,4)=2n-4$, with the upper bound witnessed by $K_{2,n-2}$; the neighbouring $k=5$ question was recently settled exactly by Fang and Hou. We prove that every graph $G$ of order $n$ and size at most $2n-10n^{2/3}-7$ has a relabelling with at most three common edges. Consequently, \[ 2n-10n^{2/3}-7<f(n,4)\leq 2n-4, \] and hence \[ f(n,4)=2n-o(n). \] Thus we resolve Erdős's four-edge intersection problem asymptotically, confirming his proposed value up to a sublinear error term. For comparison, for all sufficiently large $n$, Fang and Hou's result guarantees at most four common edges for graphs with at most $2n-3$ edges, whereas reducing the edge bound by only $10n^{2/3}+4=o(n)$ already allows us to guarantee at most three common edges.
