Research in Combinatorics

该奖项为离散数学的三个领域——极值组合学、拉姆齐理论和概率方法——的一些研究课题的调查提供资金。极值组合是组合数学的重要组成部分。例如，图论中的一个典型问题如下：给定一个图F，确定在不包含F的n阶图中边的最大数目ex （n，F）。拉姆齐理论研究在受限分区下保留的结构。自然，经典的例子是Ramsey定理，它以最简单的形式告诉我们任意大的2边彩色完全图必须包含任意大的单色完全子图。这里的划分是由着色给出的，在这个划分下保留的结构是完全图。概率方法已被证明是组合学中一种强大的技术，而随机图论是图论的一个分支，从这些方法中产生。在过去的三年里，申请人的研究主要集中在这些领域。本项目的工作将包括Ramsey理论中的一些问题，正则引理的可能扩展问题，随机图的极值性质，delta系统等问题。这项研究属于组合学领域。组合学的目标之一是找到研究离散对象集合如何排列的有效方法。离散系统的行为对现代通信极为重要。例如，大型网络的设计，比如那些出现在电话系统中的网络，以及计算机科学中处理离散对象集的算法设计，这就利用了组合研究。

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.