Extremal Graph Theory and Bootstrap Percolation

摘要：Balog dms -0302804奖提出的研究领域为极值图论和自举渗流。它们相距不远，因为在第一个中使用了许多概率工具，而在第二个中需要许多组合思想。计算机科学的需要和离散模型发挥越来越重要作用的应用需求，增加了极值图论的重要性，并提出了一种算法观点。近四十年来，渗流理论一直是概率论、组合学和物理学相结合的一个活跃的研究领域。对标准渗流的各个方面的兴趣仍然很高，包括对临界概率的估计。近年来，人们研究了越来越多的标准渗流模型的变体，特别是被称为自举渗流的过程族。最近的应用出现在不同的方面，例如时空动力系统。物理学家进行的计算机实验已经提出了有趣的非平凡的大规模行为，许多深奥的数学结果已经在许多模型中得到了证明。该方法旨在研究临界概率下的渗流过程。提议者的工作是将图兰定理扩展到几个方向。一个方向是描述不包含某些诱导子图的图族。二是研究超图上的图兰问题，特别是三重系统上的图兰问题，并开发正则性定理和稳定性定理等通用工具。自举渗透是随机元胞自动机家族中的一员，它是一个图上的过程，其中每个站点以一定的概率打开或关闭，并且这些状态随时间而变化。研究自举渗流，主要目的是描述相变，估计临界概率，以及临界概率周围窗口的大小。计划是证明转换是尖锐的，并研究不同的模型，其理解将有助于应用程序。

Abstract for award of Balog DMS-0302804The proposed research areas are extremal graph theory and bootstrap percolation. They are not far from each other, as many probabilistic tools are used in the first one, and many combinatorial ideas are needed in the second one. The need of computer science and demands from applications where discrete models play more and more important roles, increase the importance of extremal graph theory and suggests an algorithmic point of view. For about forty years now, percolation theory has been an active area of research at the interface of probability theory, combinatorics and physics. Interest in various aspects of standard percolation remains high, including estimates of critical probabilities. Lately more and more variants of the standard percolation models have been studied, in particular, the family of processes known as bootstrap percolation. Recent applications arise from different aspects, for example from spatio-temporal dynamical systems. Computer experiments performed by physicists have suggested interesting non-trivial large-scale behavior, and many deep mathematical results have been proved about a number of models.The proposer is aiming to study the percolation process at the critical probability.The work of the proposer is an extension of Turan's Theorem into several directions. One direction is to describe graph families which do not contain certain induced subgraphs. The other is to study Turan type of questions on hypergraphs, in particular on triple systems, and to develop general tools like regularity and stability theorems.Bootstrap percolation, a member of the family of random cellular automata, is a process on graphs, where each site is open or closed with a certain probability, and these states are changing with time.Studying bootstrap percolation, the main aim of the proposer is to describe the phase transition, estimate the critical probability, and the size of the window around the critical probability. The plan is to prove that the transitions are sharp, and to investigate different models, whose understanding would be helpful in the applications.