CHROMPART · Partition calculus on graphs, digraphs and hypergraphs with uncountable chromatic number

Our main goal is to develop the theory of partition calculus on graphs, digraphs and hypergraphs with emphasis on interactions between one-and multi-dimensional relations. Such global characteristics crucially depend on local, often finitary structural properties. This places our project at the meeting point of finite and infinite combinatorics with logic and set theory. Some of the most important questions that motivate our investigations were first raised by P. Erdős and A. Hajnal in the 1960s. Their problems still guide research across finite and infinite combinatorics including the most recent works of R. Diestel, N. Hindman, P. Komjáth, C. Thomassen, S. Todorcevic, and S. Shelah. Our main objective is to investigate ramification arguments between Ramsey-results of varying dimensions. In fact, (1) we study if graphs with uncountable chromatic number necessarily satisfy the same higher-dimensional negative partition relations as uncountable complete graphs. We relate this theme to (2) the existence of orientations with large dichromatic number and partition relations on digraphs. Lastly, we explore a novel concept, (3) the existence of oscillation maps on the obligatory hypergraph associated to a graph with uncountable chromatic number. Our program will be carried out through solving specific, often well-known open problems that are central to these themes. We aim to study both the purely combinatorial and the deep foundational issues that underlie these questions. Hence, we will complement the use of advanced forcing techniques from set theory (such as mixed side-condition methods and new iteration preservation theorems) with novel combinatorial tools, such as minimal walks, oscillation maps and various ZFC construction scheme techniques. We expect our research to produce new methods of wide impact and a significantly deeper understanding of the interactions of finite and infinitary combinatorics.