# Large Discrete Structures

## Periodic Reporting for period 4 - LaDIST (Large Discrete Structures)

Reporting period: 2020-03-01 to 2020-12-31

The main aim of this project is 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 are studied in this project are of importance in particular in computer science where very large networks appear frequently.

In the first half of the lifetime of the project, we particularly focused on analyzing mathematical objects that are used to represent large dense graphs, which are called graphons, and specifically graphons that are uniquely determined by finitely many density constraints. Such graphons are called finitely forcible graphons. One of the results of the project is that every graphon can be contained as a substructure (i.e. embedded) in a finitely forcible graphon. In other words, graphons that can be described in a simple finitary way can have arbitrarily complex structure. This result provides a universal framework for constructing very complex finitely forcible graphons and includes as special cases several ad hoc counterexample constructions to major conjectures in the area, which asserted that finitely forcible graphons always always have a simple structure. One of the main motivations for the study of finitely forcible graphons comes from extremal combinatorics. Indeed, Lovasz conjectured that every problem in extremal graph theory has an optimum finitely forcible solution. This problem was considered to be one of the most important problems in the area, and another result of the project is that this conjecture fails in a strong sense.

We also considered applications of the methods that we were developing, in particular, applications inside mathematics and in computer science. The methods related to representing large graphs allowed us to solve several conjectures and make substantial progress on several other open problems in mathematics, in particular in extremal combinatorics. While the project does not include developing software applications, some of the results that we have obtained are of algorithmic nature and have a relevance to computer science. In these cases, we have prepared for presentation and presented our results at appropriate computer science conferences to bring them to the attention of the computer science community.
This ERC project belongs to discrete mathematics, however, it also addresses several topics from computer science. The studied problems are concentrated along the following four main themes: structure of dense graph limits, extremal combinatorics, bridging the dense and sparse limits, and algorithmic challenges. The work on the project in the reporting period primarily focused on dense graph limits and extremal combinatorics. The full list of research papers prepared with the support of the project can be found at the dedicated website https://warwick.ac.uk/fac/sci/maths/people/staff/daniel_kral/ladist/publications/ . It should be noted that all papers supported by the project are made freely available on the preprint server arXiv at the time of submission. In the reporting period, the project results have been a subject of 15 research papers, and 2 additional papers containing results obtained in the reporting period were finalized in June and July 2018. Most of the papers have not yet been accepted for publication because of a slow speed of review process in mathematics and theoretical computer science. The project also supported two research workshops that brought together researchers from different areas of discrete mathematics and computer science - one took place in December 2016 and the other took place in June 2018, just after this reporting period concluded.
The project has already gone significantly beyond the start of the art. The particular highlight is proving the universality of finitely forcible graphons and a counterexample to a conjecture of Lovász on finite forcibility of optimal solutions of extremal graph theory problems (which was the second most often quoted open problem in the theory of graph limits). In the further stages of the project, the insights obtained in the initial period of the project will assist with developing techniques to better represent large networks/graphs and make further progress on specific open problems, in particular those from extremal combinatorics.