## Final Report Summary - EC (Extremal Combinatorics)

This project dealt with both classical areas of extremal combinatorics as well as with the recently established deep connections between discrete and analytic structures.

Some fundamental properties of the set of hypergraph Turan densities were established, solving a number of open questions posed by R.Baber, F.Chung, V.Falgas-Ravry, R.Graham, C.Grosu, J.Talbot, and E.Vaughan. For example, it was proved that there are continuum-many hypergraph Turan densities (that is, there are as “many” densities as real numbers) and that every "plausible" construction is the solution of some finite Turan problem. This progress is remarkable given that the hypergraph Turan problem is notoriously difficult. For instance, only one non-trivial value of the co-degree Turan density was known until recently; the project extended this list with two more natural examples.

A number results were obtained on estimating the number of substructures under various constraints. A notable example is an almost complete solution (apart from an asymptotically negligibly set of parameters) to the famous Erdos-Rademacher problem from 1941 on the minimum number of triangles in a graph of given order and size. Also, a substantial progress was obtained on the problem of P.Erdos from 1962 on the smallest density of k-cliques (complete graphs of order k) in a graph with independence number less than l. It is remarkable that such basic case of the problem as (k,l)=(3,4) withstood all previous attempts and was solved only recently, even though an optimal construction is very simple: 3 cliques without any edges between them. The proofs use flag algebras (a computer-based approach) and general stability-type arguments. In order to illustrate the power of these techniques, note that we can now solve cases like (k,l)=(7,3) where the optimal construction has 16 cliques with 80 complete bipartite graphs added between them. Powerful sufficient conditions for establishing the stability and exact results have been established and have been implemented in a freely available computer package flagmatic.

Some bounds in Ramsey theory were improved: namely, for Gowers' c0 theorem and the Graham-Rothschild Theorem on m-parameter words. For example, the new bounds for Gowers' c0 theorem belongs to the class E7 of Grzegorczyk's hierarchy (while no previous effective bounds were known).

A novel application of combinatorial arguments and graphings (objects that arise as limits of bounded degree graphs) led to a number of results on equidecomposability where, for given sets A and B, one has to partition A into finitely many parts that can be rearranged to form a partition of B. Most of the previous positive results crucially rely on the Axiom of Choice and therefore produce non-measurable parts in general. Here it was shown that, under certain natural assumptions, one can require that all parts are both Baire and Lebesgue measurable. In particular, measurable and translation-only versions of Tarski's circle squaring, Hilbert's third problem, and Wallace-Bolyai-Gerwien's theorem were obtained. Furthermore, a characterisation of sets measurably equidecomposable to a cube was established in dimension 3 or higher.

The project also produced some generally applicable results on limits of discrete structures as follows. Asymptotically optimal edge-colourings were constructed for an arbitrary graphing. Answering a questions of S.Janson, it was shown that a limit object for partially ordered sets can be brought to some basic form (namely, to be supported on the standard unit interval). As another example, new methods were introduced for establishing finite forcibility for permutation limits, solving R.Graham’s question whether bounded statistics can force permutations to look “random”.

One of the features of the project is that it connects combinatorics to many other areas (descriptive set theory, functional analysis, group theory, large-scale semi-definite programming, measure theory, etc). Such connections enrich all areas involved, leading to a better general understanding as well as new powerful methods and tools.

Some fundamental properties of the set of hypergraph Turan densities were established, solving a number of open questions posed by R.Baber, F.Chung, V.Falgas-Ravry, R.Graham, C.Grosu, J.Talbot, and E.Vaughan. For example, it was proved that there are continuum-many hypergraph Turan densities (that is, there are as “many” densities as real numbers) and that every "plausible" construction is the solution of some finite Turan problem. This progress is remarkable given that the hypergraph Turan problem is notoriously difficult. For instance, only one non-trivial value of the co-degree Turan density was known until recently; the project extended this list with two more natural examples.

A number results were obtained on estimating the number of substructures under various constraints. A notable example is an almost complete solution (apart from an asymptotically negligibly set of parameters) to the famous Erdos-Rademacher problem from 1941 on the minimum number of triangles in a graph of given order and size. Also, a substantial progress was obtained on the problem of P.Erdos from 1962 on the smallest density of k-cliques (complete graphs of order k) in a graph with independence number less than l. It is remarkable that such basic case of the problem as (k,l)=(3,4) withstood all previous attempts and was solved only recently, even though an optimal construction is very simple: 3 cliques without any edges between them. The proofs use flag algebras (a computer-based approach) and general stability-type arguments. In order to illustrate the power of these techniques, note that we can now solve cases like (k,l)=(7,3) where the optimal construction has 16 cliques with 80 complete bipartite graphs added between them. Powerful sufficient conditions for establishing the stability and exact results have been established and have been implemented in a freely available computer package flagmatic.

Some bounds in Ramsey theory were improved: namely, for Gowers' c0 theorem and the Graham-Rothschild Theorem on m-parameter words. For example, the new bounds for Gowers' c0 theorem belongs to the class E7 of Grzegorczyk's hierarchy (while no previous effective bounds were known).

A novel application of combinatorial arguments and graphings (objects that arise as limits of bounded degree graphs) led to a number of results on equidecomposability where, for given sets A and B, one has to partition A into finitely many parts that can be rearranged to form a partition of B. Most of the previous positive results crucially rely on the Axiom of Choice and therefore produce non-measurable parts in general. Here it was shown that, under certain natural assumptions, one can require that all parts are both Baire and Lebesgue measurable. In particular, measurable and translation-only versions of Tarski's circle squaring, Hilbert's third problem, and Wallace-Bolyai-Gerwien's theorem were obtained. Furthermore, a characterisation of sets measurably equidecomposable to a cube was established in dimension 3 or higher.

The project also produced some generally applicable results on limits of discrete structures as follows. Asymptotically optimal edge-colourings were constructed for an arbitrary graphing. Answering a questions of S.Janson, it was shown that a limit object for partially ordered sets can be brought to some basic form (namely, to be supported on the standard unit interval). As another example, new methods were introduced for establishing finite forcibility for permutation limits, solving R.Graham’s question whether bounded statistics can force permutations to look “random”.

One of the features of the project is that it connects combinatorics to many other areas (descriptive set theory, functional analysis, group theory, large-scale semi-definite programming, measure theory, etc). Such connections enrich all areas involved, leading to a better general understanding as well as new powerful methods and tools.