External Combinatorics and Codes

研究了不同组合结构的局部特性对全局参数的影响。这就是所谓图兰数问题的一般框架。研究者强调了四个不同的方面：1。研究三重系和多图的图兰数，作为实现超图一般理论的工具。2. 研究覆盖半径问题，特别是涉及图兰数自然出现的等权码。3. 研究更一般的编码理论问题，如叠加码、导致超图相交问题的识别码。4. 寻找几何和代数表示，比如洛瓦兹的香农容量界，拉马努金图，极性图，图的布拉格维，图兰数自然出现的地方。大多数有限问题都可以表述为极值图或超图问题。极值组合学应用了许多其他数学领域的工具和结果，如数论、线性和交换代数、概率论、几何和信息论。另一方面，它在组合学、几何、整数规划、计算机科学、编码理论、部分有序集的维数理论、加密等各个领域都有很多有趣的应用。极值组合学和编码理论在计算机科学和通信理论中的应用是不可或缺的。

The investigator studies how local properties affect the global parameters of different combinatorial structures. This is a very general framework of the so called Turan number problems. The investigator emphasizes four different aspects 1. To study the Turan numbers of triple systems and multigraphs, as a tool to achieve a general theory for hypergraphs. 2. To study covering radius problems, especially concerning constant weight codes where Turan numbers naturally emerge. 3. To study more general coding theory problems, like superimposed codes, identifying codes which lead to hypergraph intersection problems. 4. To find geometrical, and algebraic representations, like Lovasz' Shannon capacity bound, Ramanujan graphs, polarity graphs, Prague dimension of graphs, where Turan numbers naturally emerge.Most finite problems can be formulated as extremal graph or hypergraph problems. Extremal combinatorics applies a broad array of tools and results from other fields of mathematics like number theory, linear and commutative algebra, probability theory, geometry, and information theory. On the other hand it has a number of interesting applications in all parts of combinatorics, and in geometry, integer programming, computer science, coding theory, dimension theory of partially ordered sets, encryptions.Applications of extremal combinatorics and coding theory in computer science and communication theory are indispensable.