Recently, some studies on the fair allocation of indivisible goods notice a connection between a purely combinatorial problem called the Rainbow Cycle problem and a fairness notion known as $\efx$: assuming that the rainbow cycle number for parameter $d$ (i.e. $\rainbow(d)$) is $O(d^\beta \log^\gamma d)$, we can find a $(1-\epsilon)$-$\efx$ allocation with $O_{\epsilon}(n^{\frac{\beta}{\beta+1}}\log^{\frac{\gamma}{\beta +1}} n)$ number of discarded goods \cite{chaudhury2021improving}. The best upper bound on $\rainbow(d)$ is improved in a series of works to $O(d^4)$ \cite{chaudhury2021improving}, $O(d^{2+o(1)})$ \cite{berendsohn2022fixed}, and finally to $O(d^2)$ \cite{Akrami2022}.\footnote{We refer to the note at the end of the introduction for a short discussion on the result of \cite{Akrami2022}.} Also, via a simple observation, we have $\rainbow(d) \in \Omega(d)$ \cite{chaudhury2021improving}. In this paper, we introduce another problem in extremal combinatorics. For a parameter $\ell$, we define the rainbow path degree and denote it by $\ech(\ell)$. We show that any lower bound on $\ech(\ell)$ yields an upper bound on $\rainbow(d)$. Next, we prove that $\ech(\ell) \in \Omega(\ell^2/\log n)$ which yields an almost tight upper bound of $\rainbow(d) \in \Omega(d \log d)$. This in turn proves the existence of $(1-\epsilon)$-$\efx$ allocation with $O_{\epsilon}(\sqrt{n \log n})$ number of discarded goods. In addition, for the special case of the Rainbow Cycle problem that the edges in each part form a permutation, we improve the upper bound to $\rainbow(d) \leq 2d-4$. We leverage $\ech(\ell)$ to achieve this bound. Our conjecture is that the exact value of $\ech(\ell) $ is $ \lfloor \frac{\ell^2}{2} \rfloor -1$. We provide some experiments that support this conjecture. Assuming this conjecture is correct, we have $\rainbow(d) \in \Theta(d)$.

翻译：最近，关于不可分物品公平分配的一些研究注意到一个纯粹组合问题——“彩虹循环问题”与一种被称为$\efx$的公平概念之间的联系：假设参数$d$的彩虹循环数（即$\rainbow(d)$）为$O(d^\beta \log^\gamma d)$，则我们可以找到一种$(1-\epsilon)$-$\efx$分配，其中丢弃的商品数量为$O_{\epsilon}(n^{\frac{\beta}{\beta+1}}\log^{\frac{\gamma}{\beta +1}} n)$ \cite{chaudhury2021improving}。关于$\rainbow(d)$的最佳上界在一系列工作中被改进为$O(d^4)$ \cite{chaudhury2021improving}、$O(d^{2+o(1)})$ \cite{berendsohn2022fixed}，最终达到$O(d^2)$ \cite{Akrami2022}。\footnote{我们请读者参考引言末尾的注释，以简要讨论文献\cite{Akrami2022}的结果。}此外，通过一个简单的观察，我们有$\rainbow(d) \in \Omega(d)$ \cite{chaudhury2021improving}。在本文中，我们引入极值组合学中的另一个问题。对于参数$\ell$，我们定义彩虹路径度并将其记为$\ech(\ell)$。我们证明，关于$\ech(\ell)$的任何下界都会导出$\rainbow(d)$的一个上界。接下来，我们证明$\ech(\ell) \in \Omega(\ell^2/\log n)$，这给出了$\rainbow(d) \in \Omega(d \log d)$的一个几乎紧的上界。这进而证明了存在一种$(1-\epsilon)$-$\efx$分配，其中丢弃的商品数量为$O_{\epsilon}(\sqrt{n \log n})$。此外，针对彩虹循环问题的特殊情况（即每个部分中的边构成一个置换），我们将上界改进为$\rainbow(d) \leq 2d-4$。我们利用$\ech(\ell)$来达到这一界。我们猜想$\ech(\ell)$的精确值为$\lfloor \frac{\ell^2}{2} \rfloor -1$，并提供一些支持这一猜想的实验。假设这一猜想正确，则有$\rainbow(d) \in \Theta(d)$。