The emerging theory of graph limits exhibits an analytic perspective on graphs, showing that many important concepts and tools in graph theory and its applications can be described more naturally (and sometimes proved more easily) in analytic language. We extend the theory of graph limits to the ordered setting, presenting a limit object for dense vertex-ordered graphs, which we call an orderon. As a special case, this yields limit objects for matrices whose rows and columns are ordered, and for dynamic graphs that expand (via vertex insertions) over time. Along the way, we devise an ordered locality-preserving variant of the cut distance between ordered graphs, showing that two graphs are close with respect to this distance if and only if they are similar in terms of their ordered subgraph frequencies. We show that the space of orderons is compact with respect to this distance notion, which is key to a successful analysis of combinatorial objects through their limits. We derive several applications of the ordered limit theory in extremal combinatorics, sampling, and property testing in ordered graphs. In particular, we prove a new ordered analogue of the well-known result by Alon and Stav [RS\&A'08] on the furthest graph from a hereditary property; this is the first known result of this type in the ordered setting. Unlike the unordered regime, here the random graph model $G(n, p)$ with an ordering over the vertices is not always asymptotically the furthest from the property for some $p$. However, using our ordered limit theory, we show that random graphs generated by a stochastic block model, where the blocks are consecutive in the vertex ordering, are (approximately) the furthest. Additionally, we describe an alternative analytic proof of the ordered graph removal lemma [Alon et al., FOCS'17].

翻译：新兴的图极限理论展示了图的分析视角，表明图论及其应用中的许多重要概念和工具可以用分析语言更自然地描述（有时更易于证明）。我们将图极限理论推广到有序场景，提出了密集顶点有序图的极限对象，称为orderon。作为特例，这产生了行和列有序的矩阵极限对象，以及随时间（通过顶点插入）扩展的动态图的极限对象。在此过程中，我们设计了有序图之间切割距离的有序局部保持变体，表明两个图在此距离下接近当且仅当它们在有序子图频率上相似。我们证明了orderon空间相对于该距离概念是紧的，这对于通过极限成功分析组合对象至关重要。我们推导了有序极限理论在极值组合学、采样和有序图性质测试中的几个应用。特别地，我们证明了Alon和Stav [RS\&A'08]关于遗传性质最远图的著名结果的新有序类比；这是有序场景中此类结果的首次已知。与无序情形不同，这里顶点具有顺序的随机图模型$G(n, p)$并不总是对某些$p$渐近地最远离该性质。然而，利用我们的有序极限理论，我们表明由随机块模型生成的随机图（其中块在顶点顺序中是连续的）是（近似）最远的。此外，我们描述了有序图移除引理 [Alon等，FOCS'17]的另一种分析证明。