🤖 AI Summary
本文解决了关于无几何交叉嵌套环的问题，通过新方法证明了对于每个固定的k≥3，fk(n)的上界为O_k(n (log log n)^2 / log log log n)，从而改进了之前的结果。
📝 Abstract
Cycles $C_1,\ldots,C_k$ in a graph are called nested without geometric crossings if they are pairwise edge-disjoint, $V(C_k)\subseteq\cdots\subseteq V(C_1)$, and each pair of consecutive cycles induces the same cyclic order on the vertices of the inner cycle, up to reversal. Let $f_k(n)$ be the least number of edges that forces such a family in every $n$-vertex graph. Answering a question of Erdős for two cycles, Gil Fernández, Kim, Kim and Liu proved that $f_2(n)=O(n)$ and asked whether $f_k(n)=O_k(n)$ for every fixed $k$. Xu, Zeng and Zhang recently obtained the first general bound, $f_k(n)=O_k\bigl(n(\log n)^{k-1}(\log\log n)^{k-3}\bigr)$ for every fixed $k\ge3$.
We prove that, for every fixed $k\ge3$, \[f_k(n)=O_k\!\left(n\,\frac{(\log\log n)^2}{\log\log\log n}\right), \] so in particular $f_k(n)\le n(\log n)^{o(1)}$, where the $n$-dependent iterated-logarithmic factor has the same form for every fixed number of cycles.