Modern extremal perspectives in random graph theory

近年来，组合学、概率论及其应用之间的联系迅速发展成为一个丰富而令人兴奋的研究领域，与其他领域有许多应用和联系，如理论计算机科学或统计物理。这个界面最近的一个趋势是将图论中的不同极值结果与随机图结合起来。这一趋势的三个主要方向是：(1)获得随机图的极值结果的类似物；(2)研究密集图与随机图的并集极值问题；(3)通过分析图的随机子图来研究图如何鲁棒地满足某些性质。该项目的目的是在这些方向上取得进展，分析随机图的标准和非标准模型。特别是，我们将考虑随机摄动图的一个新方向，分析随机几何图，并广泛研究超立方体的随机子图。这些主题与计算机科学中的应用密切相关，特别是当感兴趣的网络很大时。

In recent years, the interface between combinatorics, probability and their applications has quickly developed into a rich and exciting area of research, with many applications and connections to other areas, such as theoretical computer science or statistical physics. A recent trend at this interface aims to combine different extremal results in graph theory with random graphs. Three of the main directions in this trend are the following: (1) obtaining analogues of extremal results for random graphs; (2) studying extremal problems in the union of a dense graph and a random graph, and (3) investigating how robustly graphs satisfy certain properties by analysing their random subgraphs. The aim of the project is to make progress in each of these directions, analysing both standard and non-standard models of random graphs. In particular, we will consider a new direction in randomly perturbed graphs, analyse random geometric graphs, and work extensively on random subgraphs of the hypercube. These topics are deeply related with applications in computer science, especially when the networks of interest are large.