In this paper, we study a Ramsey type problems dealing with the number of ordered subgraphs present in an arbitrary ordering of a larger graph. Our first result implies that for every vertex ordered graph G on k vertices and any stochastic vector $$\overrightarrow{a}$$a→ with k! entries, there exists a graph H with the following property: for any linear order of the vertices of H, the number of induced ordered copies of G in H is asymptotically equal to a convex combination of the entries in $$\overrightarrow{a}$$a→. This for a particular choice of $$\overrightarrow{a}$$a→ yeilds an earlier result of Angel, Lyons, and Kechris. We also consider a similar question when the ordering of vertices is replaced by the ordering of pairs of vertices. This problem is more complex problem and we prove some partial results in this case.

中文翻译：

---

订单统计

在本文中，我们研究了一个 Ramsey 类型的问题，该问题处理存在于较大图的任意排序中的有序子图的数量。我们的第一个结果意味着对于 k 个顶点上的每个顶点有序图 G 和任何随机向量 $$\overrightarrow{a}$$a→ with k! 条目，存在具有以下性质的图 H：对于 H 的顶点的任何线性顺序，H 中 G 的诱导有序副本的数量渐近等于 $$\overrightarrow{a} 中条目的凸组合$$a→。这对于 $$\overrightarrow{a}$$a→ 的特定选择产生了 Angel、Lyons 和 Kechris 的早期结果。当顶点的排序被顶点对的排序代替时，我们也考虑了一个类似的问题。这个问题是更复杂的问题，我们在这种情况下证明了一些部分结果。