## Periodic Reporting for period 2 - TurantypeProblems (Turan-type problems for graphs and hypergraphs)

Reporting period: 2020-09-30 to 2021-09-29

Advances in technology have called for the need to understand large complex networks. All networks in real world can be modeled using a graph or a hypergraph: individual objects are represented using nodes (vertices) while connections among groups of them are represented using subsets (hyperedges). For graphs, these connections are between pairs of nodes, while for hypergraphs any number of nodes can interact within each subgroup. What sort of substructures must be present in a network given we know how dense the network is? This is the general question that Turán-type problems address. In this project, we study two sets of problems; the first one is Turán-type extremal problems for graphs and hypergraphs, the second one is extremal problems in Latin squares and the existence of rainbow subgraphs in edge-coloured graphs.

In Turán-type extremal problems, roughly speaking, we want to determine how dense a system (graph, hypergraph, set system, etc.) can be when certain subconfigurations are forbidden. Turán-type problems have played a central role in the development of extremal combinatorics, as studying these problems helps us to understand how local structures and their distribution throughout a system affect global parameters of the system. This has also its real-world applications as large networks arise frequently when we model real-world human activities (the web graph modeling the internet being one example). Studying and analyzing these networks contribute significantly to the efficiency and well-functioning of modern society. Pursuing Turán-type problems will also inspire new techniques with applications throughout discrete mathematics, since we are pushing extremal combinatorics into new and challenging directions. Throughout the course of more than sixty years, efforts to tackle these problems have led to the introduction of new and powerful tools that draw ideas from other branches of mathematics such as probability, algebra, geometry, number theory, and analysis. These tools have in return greatly propelled the developments of extremal combinatorics, which has applications in other fields of mathematics, such as number theory, additive combinatorics and in computer science, such as communication complexity, information theory and others.

The second set of problems concerns Latin squares and rainbow subgraphs in graphs. An nxn Latin square is an nxn square filled with n different symbols, each of which occurs exactly once in each row and column. This definition might remind the reader of a commonly known puzzle called ‘sudoku’ which is a 9x9 Latin square with some additional constraints. The study of Latin squares is very old; it dates back to the 1700s to the work of the famous mathematician Leonard Euler. Latin squares have connections to various fields in mathematics. For example, they are multiplication tables of quasigroups from algebra. They are also used as error-correcting codes when, for example, one wishes to transmit data via a noisy channel, such as power lines. No matter how one fills a 3x3 Latin square, it is always possible to pick three cells no two of which share a row, column or symbol. Such a collection of cells is called a transversal. In contrast, it is not possible to find a transversal in a 2x2 Latin square. A famous conjecture from the 1960s due to Ryser asserts that for odd n, it is always possible to find a transversal in an nxn Latin square. In this project we study this and related problems. Progress on this question would lead to the development of new methods connecting combinatorics and algebra.

In Turán-type extremal problems, roughly speaking, we want to determine how dense a system (graph, hypergraph, set system, etc.) can be when certain subconfigurations are forbidden. Turán-type problems have played a central role in the development of extremal combinatorics, as studying these problems helps us to understand how local structures and their distribution throughout a system affect global parameters of the system. This has also its real-world applications as large networks arise frequently when we model real-world human activities (the web graph modeling the internet being one example). Studying and analyzing these networks contribute significantly to the efficiency and well-functioning of modern society. Pursuing Turán-type problems will also inspire new techniques with applications throughout discrete mathematics, since we are pushing extremal combinatorics into new and challenging directions. Throughout the course of more than sixty years, efforts to tackle these problems have led to the introduction of new and powerful tools that draw ideas from other branches of mathematics such as probability, algebra, geometry, number theory, and analysis. These tools have in return greatly propelled the developments of extremal combinatorics, which has applications in other fields of mathematics, such as number theory, additive combinatorics and in computer science, such as communication complexity, information theory and others.

The second set of problems concerns Latin squares and rainbow subgraphs in graphs. An nxn Latin square is an nxn square filled with n different symbols, each of which occurs exactly once in each row and column. This definition might remind the reader of a commonly known puzzle called ‘sudoku’ which is a 9x9 Latin square with some additional constraints. The study of Latin squares is very old; it dates back to the 1700s to the work of the famous mathematician Leonard Euler. Latin squares have connections to various fields in mathematics. For example, they are multiplication tables of quasigroups from algebra. They are also used as error-correcting codes when, for example, one wishes to transmit data via a noisy channel, such as power lines. No matter how one fills a 3x3 Latin square, it is always possible to pick three cells no two of which share a row, column or symbol. Such a collection of cells is called a transversal. In contrast, it is not possible to find a transversal in a 2x2 Latin square. A famous conjecture from the 1960s due to Ryser asserts that for odd n, it is always possible to find a transversal in an nxn Latin square. In this project we study this and related problems. Progress on this question would lead to the development of new methods connecting combinatorics and algebra.

Firstly, significant progress has been made on Ryser’s conjecture. This is a conjecture which says that every n by n Latin square has a transversal of order n-1, moreover, if n is odd, it has a full transversal. Most research towards Ryser's conjecture has focused on proving that all n by n Latin squares have large transversals, that is of order n-o(n). In scopes of this project, together with Alexey Pokrovskiy, Peter Keevash and Benny Sudakov we gave the first improvement to a forty year old result by Hatami and Shor. The improvement is in the term o(n), we managed to reduce the error o(n) from poly-logarithmic bound to sub-logarithmic bound. Our method is versatile enough to give improvements for several other problems; for example, we also improved the best known bounds on a conjecture of Brouwer from 80's on the size of a matching in a Steiner triple system, and on another problem concerning Latin rectangles.

Secondly, given two r-uniform hypergraphs G and H, the Turán number is the maximum number of edges in an H-free subgraph of G. Together with Dhruv Mubayi, we studied the ‘typical’ value of the Turan number of the r-uniform linear cycle of even length. Here typical means in the Erdos-Rényi random graph where every edge is present with probability p. We use so-called the “balanced” supersaturation of r-uniform linear cycles together with the container method which has been a powerful technique to give estimates on the number of H-free graphs on a fixed number of vertices.

The size Ramsey number asks for the smallest graph in terms of edges, for which such a monochromatic copy of a fixed graph H can be found. One of the common questions in this area is understanding for which graphs/hypergraphs the size Ramsey number is linear in terms of the number of vertices of the graph H. Together with Shoham Letzter and Alexey Pokrovskiy we showed a large a family of bounded degree graphs/hypergraphs for which the size Ramsey number is linear, which essentially includes and generalizes all currently known results about the linearity of size Ramsey numbers for graphs and hypergraphs. This family includes powers of tight paths, powers of bounded degree hypergraph trees and powers of long bounded subdivisions, in both hypergraph and graph setting.

Secondly, given two r-uniform hypergraphs G and H, the Turán number is the maximum number of edges in an H-free subgraph of G. Together with Dhruv Mubayi, we studied the ‘typical’ value of the Turan number of the r-uniform linear cycle of even length. Here typical means in the Erdos-Rényi random graph where every edge is present with probability p. We use so-called the “balanced” supersaturation of r-uniform linear cycles together with the container method which has been a powerful technique to give estimates on the number of H-free graphs on a fixed number of vertices.

The size Ramsey number asks for the smallest graph in terms of edges, for which such a monochromatic copy of a fixed graph H can be found. One of the common questions in this area is understanding for which graphs/hypergraphs the size Ramsey number is linear in terms of the number of vertices of the graph H. Together with Shoham Letzter and Alexey Pokrovskiy we showed a large a family of bounded degree graphs/hypergraphs for which the size Ramsey number is linear, which essentially includes and generalizes all currently known results about the linearity of size Ramsey numbers for graphs and hypergraphs. This family includes powers of tight paths, powers of bounded degree hypergraph trees and powers of long bounded subdivisions, in both hypergraph and graph setting.

Our approach on Ryser’s conjecture builds on a probabilistic technique called the Rödle nibble. This is a powerful technique with applications in many areas like hypergraph matching and graph decomposition. It was developed by Rödl to prove the Erdös-Hanani Conjecture on existence of approximate designs by essentially showing that regular hypergraphs with bounded co-degrees have nearly-perfect matchings. Our methods combine in a novel way the random approach together with the deterministic approach. We prove and make use of the robust expansion properties of edge-coloured pseudorandom graphs. Our method is versatile enough to give improvements for several other problems, and we think it can be extended further. Our method used to obtain the linearity of the size Ramsey numbers of graphs and hypergraphs is versatile and provides a systematic approach for studying these numbers. We prove several auxiliary Ramsey-type results which are interesting on their own. In comparison to previous results, we also have one unified proof for more than two colours case and the two colours case. The main feature of our approach for the Turán-type problems is the exploitation of supersaturation of certain substructures which is used in conjunction with local constraints to build the forbidden structure once the host graph is too dense. This approach has the potential of being developed into a general tool, for example, to tackle Turrán-type problems for bipartite graphs.