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.
