The induced size-Ramsey number \(\hat{r}_\text {ind}^k(H)\) of a graph H is the smallest number of edges a (host) graph G can have such that for any k-coloring of its edges, there exists a monochromatic copy of H which is an induced subgraph of G. In 1995, in their seminal paper, Haxell, Kohayakawa and Łuczak showed that for cycles, these numbers are linear for any constant number of colours, i.e., \(\hat{r}_\text {ind}^k(C_n)\le Cn\) for some \(C=C(k)\). The constant C comes from the use of the regularity lemma, and has a tower type dependence on k. In this paper we significantly improve these bounds, showing that \(\hat{r}_\text {ind}^k(C_n)\le O(k^{102})n\) when n is even, thus obtaining only a polynomial dependence of C on k. We also prove \(\hat{r}_\text {ind}^k(C_n)\le e^{O(k\log k)}n\) for odd n, which almost matches the lower bound of \(e^{\Omega (k)}n\). Finally, we show that the ordinary (non-induced) size-Ramsey number satisfies \(\hat{r}^k(C_n)=e^{O(k)}n\) for odd n. This substantially improves the best previous result of \(e^{O(k^2)}n\), and is best possible, up to the implied constant in the exponent. To achieve our results, we present a new host graph construction which, roughly speaking, reduces our task to finding a cycle of approximate given length in a graph with local sparsity.

图H的导出大小拉姆齐数\(\hat{r}_\text {ind}^k(H)\)是（主）图G可以具有的最小边数，使得对于任何k -对其边缘进行着色，存在H的单色副本，它是G的诱导子图。 1995 年，Haxell、Kohayakawa 和 Łuczak 在他们的开创性论文中表明，对于循环，这些数字对于任何恒定数量的颜色都是线性的，即\(\hat{r}_\text {ind}^k(C_n)\ le Cn\)对于某些\(C=C(k)\)。常数C来自正则引理的使用，并且对k具有塔类型依赖性。在本文中，我们显着改进了这些界限，表明当n为偶数时， \(\hat{r}_\text {ind}^k(C_n)\le O(k^{102})n\)，从而仅获得C对k的多项式依赖。我们还证明了奇数n的\(\hat{r}_\text {ind}^k(C_n)\le e^{O(k\log k)}n\)，它几乎匹配\( e^{\欧米茄 (k)}n\)。最后，我们表明，对于奇数n，普通（非诱导）大小拉姆齐数满足\(\hat{r}^k(C_n)=e^{O(k)}n\)。这大大改善了\(e^{O(k^2)}n\)的最佳先前结果，并且是最好的，直到指数中隐含的常数。为了实现我们的结果，我们提出了一种新的主​​图构造，粗略地说，它将我们的任务简化为在具有局部稀疏性的图中找到近似给定长度的循环。