Duality in infinite combinatorics

• Applicant: Dr. Attila Joó
• Subject Area: Mathematics
• Term: since 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 513023562

Duality is a well-understood concept in finite combinatorics. Linear programing and totally unimodular matrices provide a general framework to obtain minimax theorems. Paul Erdős observed during his school years that König's and Menger's theorem remain true in infinite graphs (by using cardinals instead of natural numbers) and he conjectured the ``right'' infinite generalisation of these theorems. He recognized that cardinality is usually an overly rough measure for extending such minimax theorems to infinite. To obtain deep and important statements, the generalisation needs to reflect the combinatorial structure instead of quantities. The proof of the Erdős-Menger conjecture due to Aharoni and Berger came several decades (and several partial results) later and is considered one of the greatest achievement in infinite graph theory. Despite of the solution of these (and a couple of more) problems, most of the related open questions remained unanswered which are the key objects of our project. Minimax theorems about packing certain paths obtained by Lovász, Cherkassky, Mader, Gallai and others are expected to admit structural extensions to infinite graphs. We are intended to settle these questions. For countable graphs we succeeded in the case of the Lovász-Cherkassky theorem. Rado asked in 1966 if it is possible to extend the concept of matroids to infinite without losing duality and minors. Based on the works of Oxley and Higgs, Diestel et al. settled Rado's problem affirmatively. Nash-Williams conjectured a generalisation of Edmonds' Matroid intersection theorem which is considered the most important open question in the field. It has been proved by the applicant for countable matroids but the general case is still open and a primary target of the proposed project.

• DFG Programme: Research Grants