The main aim of this project has been to study and develop mathematical methods to analyze and approximate large discrete structures, in particular large graphs. A graph is a mathematical model of a network - a computer network, a social network, etc. So, the methods that have been studied in this project are of importance in particular in computer science where very large networks appear frequently.
In the framework of the project, we have analyzed mathematical objects that are used to represent large dense graphs, which are called graphons, and other large discrete structures such as permutations, Latin squares, etc. One of the main results of the project is a counterexample to a conjecture of Lovász that every problem in extremal graph theory has a finitely forcible optimum solution, which was also discussed during his Abel Prize Lecture in 2021 (a graphon is finitely forcible if it is uniquely determined by finitely many density constraints). The gained insights have been used to solve specific problems in extremal combinatorics, including several long-standing open problems. Applications of the developed methods in computer science have also been sought. While the project has not included developing software applications, some of the obtained results are algorithmic nature and have relevance to computer science. In such cases, the obtained results have been presented at appropriate computer science conferences to bring them to the attention of the computer science community.
