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

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

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.

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.