Periodic Reporting for period 1 - FDC (Finite and Descriptive Combinatorics)
Reporting period: 2022-01-01 to 2023-06-30
This project will explore emerging deep connections between the areas that study finite combinatorial structures and those that study analytic objects (such as descriptive set theory, functional analysis, infinite groups, large-scale semi-definite programming, measure theory, etc), with applications going both ways.
One part of this project is to apply combinatorial methods in search of algorithms for various analytic problems whose currently known solutions rely on the Axiom of Choice (and thus are not constructive). In the other direction, many important unsolved problems of extremal combinatorics will be approached via the so-called limits of discrete structures (which are analytic objects of bounded complexity approximating combinatorial objects that are too big to deal with directly).
The project will also study some fundamental questions that are of great importance on their own in extremal combinatorics (Turan and Ramsey-type questions, quasi-randomness, etc), descriptive combinatorics (decomposing two sets into equal pieces, constructive solutions to analytic constraint satisfaction problems, etc) and the theory of limits (approximation by finite graphs, identification using partial information, etc).
While this project is theoretical in nature, many objects that it studies have potential practical applications. Graph limits give us a suite of new tools for studying, modelling and predicting real-life networks, including the cases when we have only partial information. Constructive solutions to analytic problems can sometimes be turned into new efficient distributed algorithms. Also, some optimal configurations in extremal combinatorics behave like codes, so new constructions may find applications in coding and information theory.
One part of this project is to apply combinatorial methods in search of algorithms for various analytic problems whose currently known solutions rely on the Axiom of Choice (and thus are not constructive). In the other direction, many important unsolved problems of extremal combinatorics will be approached via the so-called limits of discrete structures (which are analytic objects of bounded complexity approximating combinatorial objects that are too big to deal with directly).
The project will also study some fundamental questions that are of great importance on their own in extremal combinatorics (Turan and Ramsey-type questions, quasi-randomness, etc), descriptive combinatorics (decomposing two sets into equal pieces, constructive solutions to analytic constraint satisfaction problems, etc) and the theory of limits (approximation by finite graphs, identification using partial information, etc).
While this project is theoretical in nature, many objects that it studies have potential practical applications. Graph limits give us a suite of new tools for studying, modelling and predicting real-life networks, including the cases when we have only partial information. Constructive solutions to analytic problems can sometimes be turned into new efficient distributed algorithms. Also, some optimal configurations in extremal combinatorics behave like codes, so new constructions may find applications in coding and information theory.
On the side of finite combinatorics, the group obtained a number of results, in particular on Turan-type problems. The k-graph Turan function asks for the maximum number of k-tuples on n vertices that avoid any given forbidden configuration. For example, if one forbids 4 vertices spanning all 4 possible triples then one obtains the famous problem of P.Turan from 1941 which is still open. Although this is a central area of extremal combinatorics that witnessed decades of active research, many problems remain unsolved and we still lack some satisfactory general theory. Two pre-prints produced by the group (arXiv:2206.03948 and arXiv:2212.08636) introduced new methods for establishing instances of the Turan problem where extremal configurations have highly varied and complicated structure. For example, both methods can produce finite Turan problems with infinitely many extremal limiting constructions, which is rather surprising. Also, X.Liu and O.Pikhurko (J of Combinatorial Theory, Series B, 161 (2023) 407-416) showed that Turan densities of k-graphs (that is, the limits of the appropriately scaled Turan function) can be very complicated as real numbers, thus disproving a conjecture of C.Grosu from 2016. Despite the above "negative" results, a number of Turan problems have been resolved. As one example, S.Glock F.Joos J.Kim M.Kuhn L.Lichev and O.Pikhurko (arxiv:2209.14177 accepted by Proceedings of the AMS) solved the first open case (forbidding 6 vertices spanning at least 4 triples) of the well-known Brown-Erdos-Sos problem from 1973.
Here are two highlights of the obtained results involving analytic objects. A.Mathe J.Noel and O.Pikhurko (arXiv:2202.01412) considerably strengthened the previous results on circle squaring, where one has to split a disk in the plane into finite many pieces and rearrange them into a square. They proved that the topological boundary of the obtained pieces can have the upper Minkowski dimension less than 2, that is, the pieces are quite regular and, in principle, can be visualised. A.Grzesik D.Kral and O.Pikhurko (arXiv:2303.04041 accepted by Combinatorics, Probability and Computing) showed that any stochastic random graph model with q blocks (where we have q different types of vertices and the probability of an edge between two vertices depends on their types) can be uniquely identified by the frequencies of graphs on at most 4q^2 vertices, thus greatly improving the previously best known bound of Cq^8 by L.Lovasz from 2012.
Here are two highlights of the obtained results involving analytic objects. A.Mathe J.Noel and O.Pikhurko (arXiv:2202.01412) considerably strengthened the previous results on circle squaring, where one has to split a disk in the plane into finite many pieces and rearrange them into a square. They proved that the topological boundary of the obtained pieces can have the upper Minkowski dimension less than 2, that is, the pieces are quite regular and, in principle, can be visualised. A.Grzesik D.Kral and O.Pikhurko (arXiv:2303.04041 accepted by Combinatorics, Probability and Computing) showed that any stochastic random graph model with q blocks (where we have q different types of vertices and the probability of an edge between two vertices depends on their types) can be uniquely identified by the frequencies of graphs on at most 4q^2 vertices, thus greatly improving the previously best known bound of Cq^8 by L.Lovasz from 2012.
The group will continue their work in the main areas of the project such as extremal combinatorics (Turan function, Ramsey problems, profiles of subgraph densities, etc), limits of discrete structures (finite forcibility, soficity, flag algebra method, etc) and descriptive combinatorics (equidecomposibility, constructive solutions to locally checkable labelling problems on measurable spaces, measurable tilings, etc).
In addition to solving concrete mathematical problems, the project will explore promising connections and build new bridges between finite combinatorics, limits of discrete structures, efficient distributed algorithms, statistical physics, descriptive set theory, random processes on infinite graphs, group actions on measure spaces, etc. In addition to using established or emerging techniques, the project will look for novel ways of applying the methods and tools of one area to another. Such connections enrich all areas involved, leading to a better general understanding as well as new powerful methods and tools.
In addition to solving concrete mathematical problems, the project will explore promising connections and build new bridges between finite combinatorics, limits of discrete structures, efficient distributed algorithms, statistical physics, descriptive set theory, random processes on infinite graphs, group actions on measure spaces, etc. In addition to using established or emerging techniques, the project will look for novel ways of applying the methods and tools of one area to another. Such connections enrich all areas involved, leading to a better general understanding as well as new powerful methods and tools.