External Combinatorics and Codes

研究局部性质如何影响不同组合结构的全局参数。这是所谓的图兰数问题的一个非常一般的框架。调查员强调了四个不同的方面 1.研究三元系和重图的Turan数，作为超图一般理论的工具。 2.研究覆盖半径问题，特别是关于图兰数自然出现的等重码。 3.研究更一般的编码理论问题，如叠加码、识别导致超图交问题的码等。 4.寻找几何和代数表示，如Lovasz的Shannon容量界，Ramanujan图，极性图，图的布拉格维数，其中自然出现Turan数。大多数有限问题可以用极值图或超图问题来表示。极值组合学应用了广泛的工具和其他数学领域的结果，如数论，线性和交换代数，概率论，几何和信息论。另一方面，它在组合学的各个部分，几何学，整数规划，计算机科学，编码理论，偏序集的维数理论，代数学中有许多有趣的应用。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必不可少的。

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.