Infinitary Ramsey Theory

该提案包括四个主要方向的研究计划。在过去的工作中，研究者证明了关于真嵌入划分关系的最强已知定理。对于每一个有限图H和每一个基数图t，都有一个用t种颜色“箭头”的图G。最近的结果给了我们一些希望，将这一结果推广到可数目标图，并发展了具有无穷多类的划分关系的“nesetri - rodl”型Ramsey理论。第二个方向是利用与达特茅斯学院的詹姆斯·鲍姆加特纳教授共同开发的一种名为“双分支”的影响深远的新方法来研究极化分配关系。第三个方向是继续与匈牙利布达佩斯Eotvos Lorand大学的Peter Komjath教授的成功合作。我们将研究划分关系的扩展——在非完全图被划分的情况下，我们将发展图和集合系统同时色数的理论。这将涉及强制方法和新的证明技术。第四个方向是继续与匈牙利科学院数学研究所的Istvan Juhasz教授的长期合作。具体目标将是研究破坏空间拓扑的分区，以及研究CCC结构的不变量。组合问题和方法已被证明是激发集理论新发展的有力工具，而新的元数学方法已导致集理论及其在集理论拓扑中的应用取得了惊人的进展。该项目旨在加强这一富有成果的合作。

The proposal consists of research plans going in four main directions. In a past work, the investigator proved the strongest known theorem about true embedding partition relations. For every finite graph H and for every cardinal t there is a graph G "arrowing" H with t colors. Recent results give some hope of extending this result to countable target graphs and developing a "Nesetril-Rodl" type Ramsey theory for partition relations with infinitely many classes. The second direction is the investigation of polarized partition relations using a far reaching new method called "double ramification" developed jointly with Professor James Baumgartner of Dartmouth College. The third direction is a continuation of a successful collaboration with Professor Peter Komjath of the Eotvos Lorand University of Budapest, Hungary. Extensions of partition relations will be investigated- for the case in which noncomplete graphs are partitioned, a theory of simultaneous chromatic number of graphs and set systems will be developed. Theqe will involve forcing methods and new proof techniques. The fourth direction is a continuation of a long standing collaboration with Professor Istvan Juhasz of the Mathematical Institute of the Hungarian Academy of Sciences. Specific targets will be investigations of partitions destroying the topology of a space, and study of invariants of CCC structures. Combinatorial problems and methods have proved to be a powerful tool in inspiring new developments in set theory, and new metamathematical methods have led to spectacular advances in set theory and its applications to set theoretical topology. The project plans to enhance this fruitful cooperation.