The celebrated Erdős–Stone–Simonovits theorem characterizes the asymptotic maximum edge density in $\mathcal{F}$-free graphs as $1-1/(\chi (\mathcal{F})-1)+o(1)$, where $\chi (\mathcal{F})$ is the minimum chromatic number of a graph in $\mathcal{F}$. In [6, Examples 25 and 31], it was shown that this result can be extended to the general setting of graphs with extra structure: the asymptotic maximum edge density of a graph with extra structure without some induced subgraphs is $1-1/(\chi (I)-1)+o(1)$ for an appropriately defined abstract chromatic number $\chi (I)$. As the name suggests, the original formula for the abstract chromatic number is so abstract that its (algorithmic) computability was left open.

In this paper, we both extend this result to characterize maximum asymptotic density of t-cliques in graphs with extra structure without some induced subgraphs in terms of $\chi (I)$ and we present a more concrete formula for $\chi (I)$ that allows us to show its computability when both the extra structure and the forbidden subgraphs can be described by a finitely axiomatizable universal first-order theory. Our alternative formula for $\chi (I)$ makes use of a partite version of Ramsey's Theorem for structures on first-order relational languages.

著名的 Erdős-Stone-Simonovits 定理描述了渐近最大边缘密度 $\mathcal{F}$- 自由图为 $1-1/(\chi (\mathcal{F})-1)+○(1)$， 在哪里 $\chi (\mathcal{F})$ 是图的最小色数 $\mathcal{F}$. 在[6，示例 25 和 31] 中，表明该结果可以扩展到具有额外结构的图的一般设置：具有额外结构的图的渐近最大边密度没有一些诱导子图是$1-1/(\chi (\mathrm{一世})-1)+○(1)$对于适当定义的抽象色数 $\chi (\mathrm{一世})$. 顾名思义，抽象色数的原始公式是如此抽象，以至于它的（算法）可计算性是开放的。

在本文中，我们都将此结果扩展为表征具有额外结构的图中t -团的最大渐近密度，而没有一些诱导子图$\chi (\mathrm{一世})$ 我们提出了一个更具体的公式 $\chi (\mathrm{一世})$当额外结构和禁止子图都可以用有限公理化的通用一阶理论来描述时，这使我们能够展示其可计算性。我们的替代公式$\chi (\mathrm{一世})$ 将拉姆齐定理的部分版本用于一阶关系语言的结构。