Ramsey-type problems in random and in randomly perturbed discrete structures, and their sharp thresholds

• Applicant: Professor Dr. Yury Person
• Subject Area: Mathematics
• Term: since 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 447645533

The study of random discrete structures and their interplay with deterministic counterparts is an active stream of research within extremal and probabilistic combinatorics. It has been observed by several researchers that the combination of a deterministic object with a random structure often leads to much better results. Bohman, Frieze and Martin introduced in 2001 the randomly perturbed graph model and proved that adding to a graph of small linear degree linearly many edges at random results very likely in a Hamiltonian graph. In the recent few years the study of randomly perturbed models has been extended to hypergraphs and sets of integers and is now an active research direction with many interesting insights. The purpose of this project is to further advance this area, especially focusing on Ramsey-type problems around randomly perturbed discrete structures (graphs, hypergraphs, sets of integers) and their relation to the corresponding statements in purely random models. Moreover, the phenomenon of the so-called sharp thresholds (where the structure undergoes a dramatic change of its characteristic properties) should be studied and Ramsey-type problems appear to be a challenging testbed.

• DFG Programme: Research Grants