Research in Combinatorics

RODL 9704114该奖项为离散数学的三个领域--极值组合数学、拉姆齐理论和概率方法--的一些研究课题提供资金。极值组合学是组合数学的重要组成部分。例如，图论中的一个典型问题是：给定一个图F，确定一个不包含F的n阶图的最大边数EX(n，F)。Ramsey理论研究了在受限划分下保持的结构。自然，经典的例子是Ramsey的定理，它的最简单形式告诉我们，任意大的2-边色完全图一定包含任意大的单色完全子图。这里的划分是由着色给出的，在该划分下保留的结构是完整的图。概率方法已被证明是组合数学中的一种强有力的方法，而随机图理论是从这些方法中产生的图论的一个分支。近三年来，笔者的研究主要集中在这几个方面。这个项目的工作将包括Ramsey理论中的一些问题，涉及正则性引理的可能扩展的问题，随机图的极值性质，关于Delta系统的问题等等。这项研究属于组合学领域。组合学的目标之一是找到有效的方法来研究离散的对象集合如何排列。离散系统的行为对于现代通信来说是极其重要的。例如，大型网络的设计，如那些发生在电话系统中的网络，以及计算机科学中的算法设计，都涉及离散的对象集，这利用了组合研究。

Rodl 9704114 This award provides funds for an investigation into some research topics in three areas of discrete mathematics - Extremal Combinatorics, Ramsey Theory and Probabilistic Methods. Extremal combinatorics is an important part of combinatorial mathematics. For example a typical problem in graph theory is as follows: given a graph F determine the maximum number ex (n,F) of edges in a graph of order n not containing F. Ramsey theory investigates structures that are preserved under restricted partitions. Naturally, the classical example is the theorem of Ramsey that, in its simplest form, tells us that arbitrarily large 2-edge-colored complete graphs must contain arbitrarily large monochromatic complete subgraphs. Here the partition is given by the coloring and the structures that are preserved under this partition are the complete graphs. Probabilistic Methods have proved to be a powerful technique in combinatorics, and the theory of random graphs is a branch of graph theory that has emerged from these methods. Over the last three years, the proposer's research was mainly concentrated within these areas. The work in this project will include some problems in Ramsey theory, problems involved with possible extensions of the regularity lemma, extremal properties of random graphs, problems on delta systems and others. This research is in the area of combinatorics. One of the goals of combinatorics is to find efficient methods of studying how discrete collections of objects can be arranged. The behavior of discrete systems is extremely important to modern communications. For example, the design of large networks, such as those occurring in telephone systems, and the design of algorithms in computer science deal with discrete sets of objects, and this makes use of combinatorial research.