AI 中文总结
本文提出基于传播的方法，解决奇洞小工具猜想并证明随机正则图中最大洞的阶，核心是构造辅助对象控制额外边。
AI 中文摘要
我们发展了一种基于传播的方法来寻找诱导环，并将其应用于两个问题。首先，我们通过构造一个具有 $e^{O(k)}$ 条边的图，其每个 $k$-边染色都包含一个长度为 $O(\log k)$ 的单色诱导奇环，解决了 Bradač、Draganić 和 Sudakov 的奇洞小工具猜想。作为推论，对于每个 $k\ge2$ 和每个足够大的奇数 $n$，有 \\\\[ \widehat R_{\mathrm{ind}}(C_n;k)=e^{\Theta(k)}n。\\\\] 证明使用了传播概率权重以及超图容器。其次，我们证明对于每个足够大的固定 $d$，随机 $d$-正则图 $G_{n,d}$ 中最大的洞以高概率具有阶 $\Theta(n\log d/d)$，解决了 Frieze 的一个问题。虽然这两个证明使用了不同的机制，但两者都从一个分布良好的辅助对象开始，并利用它来控制可能破坏诱导性的额外边。
英文摘要
We develop a spread-based approach to finding induced cycles and apply it to two problems. First, we resolve the odd-hole gadget conjecture of Bradač, Draganić and Sudakov by constructing an $e^{O(k)}$-edge graph whose every $k$-edge-colouring contains a monochromatic induced odd cycle of length $O(\log k)$. As a consequence, for every $k\ge2$ and every sufficiently large odd $n$, $$ \widehat R_{\mathrm{ind}}(C_n;k)=e^{Θ(k)}n. $$ The proof uses spread probability weights together with hypergraph containers. Second, we prove that for every sufficiently large fixed $d$, with high probability the largest hole in the random $d$-regular graph $G_{n,d}$ has order $Θ(n\log d/d)$, resolving a problem of Frieze. Although the two proofs use different mechanisms, both begin with a well-distributed auxiliary object and use it to control the extra edges that could destroy inducedness.
