Periodic Reporting for period 2 - FDC (Finite and Descriptive Combinatorics)
Reporting period: 2023-07-01 to 2024-12-31
This project explores 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 large number of results, in particular, on the 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. 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.
One of the most important new results obtained in this direction appeared in the recent paper by the PI (Advances in Mathematics, 464 (2025) 110148) that provided constructions of Turan (k+R,k)-systems (that is, for the Turan problem when we forbid the complete k-graph on k+R vertices) that have the optimal size within a constant factor as R is fixed and k tends to infinity. This disproved the famous and long-standing question of de Caen from the early 1990s (who offered 500 Canadian dollars for its resolution).
Two other papers (co)authored by members of the group, namely by J.Hou H.Li X.Liu D.Mubayi and Y.Zhang (Discrete Analysis 2023:18, 34 pp) and by X.Liu and O.Pikhurko (arXiv:2212.08636) introduced new general 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. For example, two papers written in collaboration of the PI and the hired PhD student Shumin Sun with S.Glock F.Joos J.Kim M.Kuhn L.Lichev (Proceedings of the American Mathematical Society Series B, 11 (2024) 173-186 and arxiv:2403.04474) solved the cases m=4,5,6,7,9 of the well-known (m+2,m)-problem of Brown-Erdos-Sos from 1973 to find the maximal number of triples on n vertices such that every set of size m+2 contains less than m triples. Previously, the case m=2 was solved in the original paper of Brown-Erdos-Sos while the case m=3 was solved by Glock in 2019.
Below are two selected obtained results that involve 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 to form a square. The question whether this is possible (with arbitrary pieces) was first raised by Tarski in 1926 motivated by the famous Banach-Tarski Paradox from 1924 that one can create two balls from one in the 3-dimensional space. The circle squaring question of Tarski remained open until it was resolved by Laczkovich in 1990. However, the proof of Laczkovich crucially relied on the Axiom of Choice, an assumption in set theory that leads to counter-intuitive and striking results (such as the Banach-Tarski Paradox). The proofs that rely on the Axiom of Choice are regarded as non-constructive from the descriptive set theory point of view and one of the key objectives of this field is to reduce or eliminate completely the use of the Axiom of Choice. It was unclear if this is possible for the circle-squaring problem until this was achieved by Grabowski-Mathe-Pikhurko (2019) and Marks-Unger (2019). These results were greatly strengthened by Mathe-Noel-Pikhurko who additionally ensured that the pieces are "simple", namely they have very low Borel complexity and their topological boundaries have small Minkowski dimension.
Motivated by old questions of Mycielski and Wagon on measurable divisibility of spheres, Grebik-Ikenmeyer-Pikhurko (arXiv:2501.13522) proved that divisibility is impossible (even in the much weaker fractional sense) if at least half of rotations are generic.
A.Grzesik D.Kral and O.Pikhurko (Combinatorics, Probability and Computing, 33 (2024) 16-31) 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 densities of graphs on at most 4q^2 vertices, thus greatly improving the previously best known bound of Cq^8 by L.Lovasz from 2012. Block graph models play a very important role in the limit theory of dense graphs, as they can be used to approximate any large graph.
One of the most important new results obtained in this direction appeared in the recent paper by the PI (Advances in Mathematics, 464 (2025) 110148) that provided constructions of Turan (k+R,k)-systems (that is, for the Turan problem when we forbid the complete k-graph on k+R vertices) that have the optimal size within a constant factor as R is fixed and k tends to infinity. This disproved the famous and long-standing question of de Caen from the early 1990s (who offered 500 Canadian dollars for its resolution).
Two other papers (co)authored by members of the group, namely by J.Hou H.Li X.Liu D.Mubayi and Y.Zhang (Discrete Analysis 2023:18, 34 pp) and by X.Liu and O.Pikhurko (arXiv:2212.08636) introduced new general 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. For example, two papers written in collaboration of the PI and the hired PhD student Shumin Sun with S.Glock F.Joos J.Kim M.Kuhn L.Lichev (Proceedings of the American Mathematical Society Series B, 11 (2024) 173-186 and arxiv:2403.04474) solved the cases m=4,5,6,7,9 of the well-known (m+2,m)-problem of Brown-Erdos-Sos from 1973 to find the maximal number of triples on n vertices such that every set of size m+2 contains less than m triples. Previously, the case m=2 was solved in the original paper of Brown-Erdos-Sos while the case m=3 was solved by Glock in 2019.
Below are two selected obtained results that involve 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 to form a square. The question whether this is possible (with arbitrary pieces) was first raised by Tarski in 1926 motivated by the famous Banach-Tarski Paradox from 1924 that one can create two balls from one in the 3-dimensional space. The circle squaring question of Tarski remained open until it was resolved by Laczkovich in 1990. However, the proof of Laczkovich crucially relied on the Axiom of Choice, an assumption in set theory that leads to counter-intuitive and striking results (such as the Banach-Tarski Paradox). The proofs that rely on the Axiom of Choice are regarded as non-constructive from the descriptive set theory point of view and one of the key objectives of this field is to reduce or eliminate completely the use of the Axiom of Choice. It was unclear if this is possible for the circle-squaring problem until this was achieved by Grabowski-Mathe-Pikhurko (2019) and Marks-Unger (2019). These results were greatly strengthened by Mathe-Noel-Pikhurko who additionally ensured that the pieces are "simple", namely they have very low Borel complexity and their topological boundaries have small Minkowski dimension.
Motivated by old questions of Mycielski and Wagon on measurable divisibility of spheres, Grebik-Ikenmeyer-Pikhurko (arXiv:2501.13522) proved that divisibility is impossible (even in the much weaker fractional sense) if at least half of rotations are generic.
A.Grzesik D.Kral and O.Pikhurko (Combinatorics, Probability and Computing, 33 (2024) 16-31) 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 densities of graphs on at most 4q^2 vertices, thus greatly improving the previously best known bound of Cq^8 by L.Lovasz from 2012. Block graph models play a very important role in the limit theory of dense graphs, as they can be used to approximate any large graph.
The group will continue their work in the main areas of the project such as extremal combinatorics (Turan function, Ramsey-type 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. While the project will use known techniques, the group will also 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 to new powerful proof techniques.
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. While the project will use known techniques, the group will also 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 to new powerful proof techniques.