EC · Extremal Combinatorics

A typical problem of Extremal Combinatorics is to maximise or minimise a certain parameter given some combinatorial restrictions. This area experienced a remarkable growth in the last few decades, having a wide range of applications that include results in number theory, algebra, geometry, logic, information theory, and theoretical computer science. There are also many practical fields that were greatly influenced by ideas from Extremal Combinatorics such as, for example, analysis of large networks, ranking of web-pages, or shotgun cloning of DNA fragments.The Principal Investigator (PI for short) will work on a number of extremal problems, with the main directions being the Tur\'an function (maximising the size of a hypergraph without some fixed forbidden subgraphs), the Rademacher-Tur\'an problem (minimising the density of F-subgraphs given the edge density), and Ramsey numbers (quantitative bounds on the maximum size of a monochromatic substructure that exists for every colouring). These are fundamental and general questions that go back at least as far as the 1940s but remain wide open despite decades of active attempts. During attacks on these notoriously difficult problems, mathematicians developed a number of powerful general methods. PI will work on extending and sharpening these techniques as well as on finding ways of applying the recently introduced concepts of (hyper)graph limits and flag algebras to concrete extremal problems. Since these concepts deal with some approximation to the studied problem, one important aspect of the project is to develop methods for obtaining exact results from asymptotic calculations (for example, via the stability approach).The support by means of a 5-year research grant will enable PI to consolidate his research and build a group in Extremal Combinatorics.