Infinitary Ramsey Theory

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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.