论文标题
彩虹Turán的$ P_5 $
The rainbow Turán number of $P_5$
论文作者
论文摘要
储层计算是预测湍流的有力工具，其简单的架构具有处理大型系统的计算效率。然而，其实现通常需要完整的状态向量测量和系统非线性知识。我们使用非线性投影函数将系统测量扩展到高维空间，然后将其输入到储层中以获得预测。我们展示了这种储层计算网络在时空混沌系统上的应用，该系统模拟了湍流的若干特征。我们表明，使用径向基函数作为非线性投影器，即使只有部分观测并且不知道控制方程，也能稳健地捕捉复杂的系统非线性。最后，我们表明，当测量稀疏、不完整且带有噪声，甚至控制方程变得不准确时，我们的网络仍然可以产生相当准确的预测，从而为实际湍流系统的无模型预测铺平了道路。
An edge-colored graph $F$ is rainbow if each edge of $F$ has a unique color. The rainbow Turán number $ex^*(n,F)$ of a graph $F$ is the maximum possible number of edges in a properly edge-colored $n$-vertex graph with no rainbow copy of $F$. The study of rainbow Turán numbers was introduced by Keevash, Mubayi, Sudakov, and Verstraëte in 2007. In this paper we focus on $ex^*(n,P_5)$. While several recent papers have investigated rainbow Turán numbers for $\ell$-edge paths $P_{\ell}$, exact results have only been obtained for $\ell < 5$, and $P_5$ represents one of the smallest cases left open in rainbow Turán theory. In this paper, we prove that $ex^*(n,P_5) \leq \frac{5n}{2}$. Combined with a lower-bound construction due to Johnston and Rombach, this result shows that $ex^*(n,P_5) = \frac{5n}{2} $ when $n$ is divisible by $16$, thereby settling the question asymptotically for all $n$. In addition, this result strengthens the conjecture that $ex^*(n,P_{\ell}) = \frac{\ell}{2}n + O(1)$ for all $\ell \geq 3$.
