奇围长的Erdős-Hajnal猜想
Locally bipartite subgraphs via multicolor Ramsey numbers
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中文总结 AI 辅助
该研究针对Erdős-Hajnal猜想，改进了$f_4(k)$的上界，还证明了奇数$g\ge5$时对应函数$h_g(k)$的增长性，证实了Mohar和Wu2018年的相关猜想。
中文摘要 AI 辅助
Erdős和Hajnal于1969年提出的著名猜想指出，对于每个整数$g\ge 4$，存在一个（最小的）函数$f_g:\mathbb{N}\rightarrow \mathbb{N}$，使得每个色数至少为$f_g(k)$的图都包含一个子图，该子图的色数至少为$k$且围长至少为$g$。迄今为止，仅Rödl在1977年证明了$g=4$的情况，对于所有$g\ge 5$，该猜想仍未解决。Rödl的精妙证明给出了$f_4(k)$的上界，是高度为$\Theta(k^2\log k)$的$k$层塔函数，这引发了改进这一巨大上界的问题。我们基于OpenAI近期关于三角形多色拉姆齐数的下界，推导出单指数上界$$f_4(k)\le e^{k^{3+o(1)}}$$。利用同系列论文中关于奇圈多色拉姆齐数的推广结果，我们证明对于每个奇数$g\ge 5$，存在一个函数$h_g:\mathbb{N}\rightarrow \mathbb{N}$，其增长速度最多为高度$\frac{g-3}{2}$的幂塔，使得每个色数至少为$h_g(k)$的图都包含一个子图，该子图的色数至少为$k$且奇围长至少为$g$，这一结果证明了Mohar和Wu于2018年提出的猜想。
英文摘要
A famous conjecture of Erdős and Hajnal (1969) states that for every integer $g\ge 4$ there is a smallest function $f_g:\mathbb{N}\to\mathbb{N}$ such that every graph of chromatic number at least $f_g(k)$ contains a subgraph of chromatic number $k$ and girth at least $g$. So far, this has only been proved for $g=4$ by Rödl (1977), with $f_4(k)$ bounded by a tower of height $Θ(k^2\log k)$. We exhibit a surprising connection between finding high-chromatic subgraphs of large odd-girth (avoiding short odd cycles) and lower-bounding multicolor Ramsey numbers of odd cycles. Using this connection, we prove that for every odd $g\ge 5$ there is a function $h_g:\mathbb{N}\to\mathbb{N}$ growing as a power tower of height $\frac{g-3}{2}$ such that every graph of chromatic number at least $h_g(k)$ contains a subgraph of chromatic number at least $k$ and odd-girth at least $g$. This proves a conjecture of Mohar and Wu (2018), addresses a question of Erdős and Hajnal (1975), and for $g=5$ improves Rödl's bound on $f_4(k)$ to a single-exponential. We extend this to a much more general meta-theorem which applies to many graph parameters: if $f$ is the fractional chromatic number, the Hall ratio, or the strict vector chromatic number (Lovász-Theta-function of the complement), then for every $k,g\in\mathbb{N}$, every graph $G$ with sufficiently large $f(G)$ contains a subgraph $G'$ of odd-girth at least $g$ with $f(G')\ge k$. The key Ramsey-theoretic ingredient is a new lower bound on Ramsey numbers of odd cycles. For $p\ge 1$, let $\mathcal{O}_p=\{C_3,C_5,\ldots,C_{2p+1}\}$. We show that $R_k(\mathcal{O}_p)\ge(\log^{(p-1)}k)^{k/3-o(k)}$ for every fixed $p$, where $\log^{(p-1)}$ denotes the $(p-1)$-fold iterated logarithm. This yields the first superexponential lower bound on multicolor Ramsey numbers of fixed odd cycles, and extends the recent breakthrough by OpenAI for triangles.
发表机构
- Department of Mathematics, ETH Zürich（苏黎世联邦理工学院数学系）
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