Problems in Probabilistic Combinatorics

1.该项目涵盖了ProbabilisticCombinatorics中的几个主题。 第一组问题是关于图的着色，主要研究满足某些局部条件的图的色数和选择数。研究者将研究的一个开放问题是最大度为d的无固定图H的复制的无固定图G的色数的界。这个问题的一个变种已经提出了二十年前由Komlos和Szemeredi，到目前为止，它只解决了一些specialcases。研究者还计划研究图的点列表染色和无圈边染色的一些相关问题，另一组问题是关于随机图和随机正则图的渐近性质。研究者的目标是了解稀疏随机图中接近最优大小的独立集的分布，并利用它来确定其选择数的渐近行为。有时候，由概率方法提供的存在性证明是不够的，最好有一个明确的结构。这一领域的主要开放问题之一是构造指数大图，没有团或大小为k的独立集。在这个项目中，研究者打算考虑这个问题连同它的双边版本。他还计划研究伪随机图的性质及其应用.不放过任何机会。这种陈词滥调体现了一种普遍的信念，即随机性在精心设计的方法论中没有地位。至少在现代组合数学中，没有什么比这更远离真理了。在这里，概率方法得到了广泛的发展，并成为最强大和最广泛使用的工具之一。概率的使用被证明有助于解决许多长期存在的开放问题。这种方法发展的另一个主要原因是随机性在理论计算机科学中的重要作用。这里的算法在其执行过程中进行随机选择，被证明是最简单和最快的许多应用程序。

1. The proposed project covers several topics in ProbabilisticCombinatorics. The first group of questions is about graph colorings.It mainly deals with the study of chromatic and choice numbers ofgraphs satisfying certain local conditions. One of the open problemwhich investigators will study is bounding the chromatic number of agraph G with maximum degree d, which contains no copy of a fixed graphH. A variant of this problem was posed already twenty years ago byKomlos and Szemeredi and so far it was solved only for some specialcases. Investigator also plans to work on a few related questionsabout vertex list coloring and acyclic edge coloring of graphs.Another set of questions is about asymptotic properties of random graphsand random regular graphs. Here investigators goal is to understandthe distribution of independent sets of nearly optimal size in sparserandom graphs, and use this to determine the asymptotic behavior ofits choice number. It is sometimes the case that an existence proof,supplied by the probabilistic method, is not sufficient and it isbetter to have an explicit construction. One of the major openproblems in this field is to construct exponentially large graphs,without a clique or an independent set of size k. In this projectinvestigator intends to consider this problem together with itsbipartite version. He also plans to study the properties ofpseudo-random graphs and their applications.2. Leave nothing to chance. This cliche embodies the common beliefthat randomness has no place in carefully planned methodologies.In modern Combinatorics at least, nothing can be further from thetruth. Here the Probabilistic Method has been developed intensivelyand become one of the most powerful and widely used tools.Use of probability proved to be helpful in tackling many longstanding open problems. Another major reason for the development ofthis method is the important role of randomness in TheoreticalComputer Science. Here algorithm which make random choices during itsexecution proved to be simplest and fastest for many applications.