发表机构
Sichuan University; Jiangsu University; Zhejiang University（四川大学; 江苏大学; 浙江大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明奇圈 $C_{2k-1}$ 的同态阈值严格低于 $\frac{1}{2k-1}$，通过新构造改进下界、新结构论证改进上界，并解决 Fox 和 Wigderson 的两个猜想。
AI 中文摘要
图 $H$ 的同态阈值 $\delta_{\text{hom}}(H)$ 询问：一个 $H$-free 图的最小度需要多大才能迫使它同态到一个有界的 $H$-free 图。确定这个阈值通常非常困难，而奇圈 $C_{2k-1}$ 是 $k\ge 3$ 时最重要的开放情形之一。Ebsen 和 Schacht 证明了上界 $\delta_{\text{hom}}(C_{2k-1})\le\frac{1}{2k-1}$，而 Sankar 利用拓扑方法和同伦等价的图论类比取得了突破，给出了第一个正的下界。值 $\frac{1}{2k-1}$ 看起来特别有说服力：Ebsen 和 Schacht 在禁止所有长度至多 $2k-1$ 的奇圈时得到了相同的精确阈值，Huang、Liu、Rong 和 Xu 后来证明了它是 $C_{2k-1}$ 的精确 blowup 阈值，Letzter 和 Snyder 也明确询问 $\delta_{\text{hom}}(C_{5})=\frac{1}{5}$ 是否成立。令人惊讶的是，我们证明上界可以被改进。更精确地，对每个整数 $k\ge3$，我们证明 $$ \frac{1}{2\left((k-1)^{4k-5}(2k-1)+\frac{(k-1)^{4k-5}-1}{k-2}\right)} \le \delta_{\text{hom}}(C_{2k-1}) \le \frac{4(k-1)}{4(k-1)(2k-1)+1} < \frac{1}{2k-1}. $$ 新的下界来自基于 Nešetřil 和 Zhu 的稀疏同态定理的一个新图论构造，并且改进了 Sankar 的定量界。改进的上界来自一个新的结构论证，该论证控制沿短奇路径的公共邻域。我们的结果有多种推论，特别是，每个长度至少为 5 的奇圈具有两两不同的色数阈值、同态阈值、多项式移除阈值和线性移除阈值，解决了 Fox 和 Wigderson 的两个猜想。
英文摘要
The homomorphism threshold $δ_{\text{hom}}(H)$ of a graph $H$ asks how large the minimum degree of an $H$-free graph has to be in order to force a homomorphism to a bounded $H$-free graph. Determining this threshold is in general very difficult, and the odd cycles $C_{2k-1}$ are among the most important open cases for $k\ge 3$. Ebsen and Schacht proved the general upper bound $δ_{\text{hom}}(C_{2k-1})\le\frac{1}{2k-1}$, while a breakthrough of Sankar, using topological methods and a graph-theoretic analogue of homotopy equivalence, gave the first positive lower bound. The value $\frac{1}{2k-1}$ appeared particularly compelling: Ebsen and Schacht obtained the same exact threshold when all odd cycles of length at most $2k-1$ are forbidden, Huang, Liu, Rong and Xu later proved that it is the exact blowup threshold of $C_{2k-1}$, and Letzter and Snyder also explicitly asked whether $δ_{\text{hom}}(C_{5})=\frac{1}{5}$. Surprisingly, we show that the upper bound can be improved. More precisely, for every integer $k\ge3$, we prove $$ \frac{1}{2\left((k-1)^{4k-5}(2k-1)+\frac{(k-1)^{4k-5}-1}{k-2}\right)} \le δ_{\text{hom}}(C_{2k-1}) \le \frac{4(k-1)}{4(k-1)(2k-1)+1} < \frac{1}{2k-1}. $$ The new lower bound comes from a new graph-theoretic construction based on a sparse homomorphism theorem of Nešetřil and Zhu, and it improves Sankar's quantitative bound. The improved upper bound follows from a new structural argument that controls common neighborhoods along short odd paths. Our results have various consequences, in particular, every odd cycle of length at least five has pairwise distinct chromatic, homomorphism, polynomial removal, and linear removal thresholds, resolving two conjectures of Fox and Wigderson.
Comments17 pages
