Periodic Reporting for period 1 - SREACEP (Sparse Robust Expander with Applications in Combinatorial Embedding Problems)
Reporting period: 2018-01-01 to 2019-12-31
One of the major developments in combinatorics in the past four decades is that of expanders. It was first introduced to construct networks (represented by graphs) that are economical (sparse) and robust (highly connected). Expanders can be defined in various different ways. From the algebraic point of view, expanders are the graphs with large spectral gap; from the probabilistic point of view, random walks on expanders are rapidly mixing; and from the graph theoretic point of view, an expander is a graph whose vertex subsets have `large' boundaries. The main purpose of this project is to study expanders from the graph theoretic point of view through its expansion property, and investigate the interplay between expanders and two other central notions in modern combinatorics: Hamilton cycles and topological minors. The results and methods related to the cycle structure in a network (graph) can be adapted and used to evaluates the network's liability to deal with failures. Also, the techniques and methods learnt and developed during these projects opened new research directions. During the fellowship, the researcher explored some of these new directions.
The researcher achieved all objectives and milestones related to Project 1 before the start of the fellowship mainly due to competition. In addition, the techniques and methods learnt and developed during this project opened new research directions. During the fellowship, the researcher explored some of these new directions. In particular, she started three projects, with different groups of collaborators, where they applied and further developed the techniques from Project 1.
In particular, the experience from WP1.1 gave the her the opportunity to work on another enumeration type problem, that is, maximising or minimising the number of substructures in a family of combinatorial object. Also, the training from WP1.2 and 1.3 familiarised the researcher with the basics and crucial tools for understanding two closely related topics: theory of graph limits and method of flag algebras. The results of these connections and training led to three projects. Two of which are on two different aspects of an extremal graph theoretic problem known as Erdős-Rademacher. Another one is related to the method of flag algebras, another theory that is closely related to both graph limits and the regularity method of Szemerédi. The last one is developing a limit theory of Latin squares.
Project 2 turned out to be out of reach, even though the researcher spent many months on it, including reading, learning, and discussing with collaborators and colleagues. However, the researcher used this opportunity to study further the related concepts and methods (embedding problems in complexes and hypergraphs) in more detail and depth.
In particular, the experience from WP1.1 gave the her the opportunity to work on another enumeration type problem, that is, maximising or minimising the number of substructures in a family of combinatorial object. Also, the training from WP1.2 and 1.3 familiarised the researcher with the basics and crucial tools for understanding two closely related topics: theory of graph limits and method of flag algebras. The results of these connections and training led to three projects. Two of which are on two different aspects of an extremal graph theoretic problem known as Erdős-Rademacher. Another one is related to the method of flag algebras, another theory that is closely related to both graph limits and the regularity method of Szemerédi. The last one is developing a limit theory of Latin squares.
Project 2 turned out to be out of reach, even though the researcher spent many months on it, including reading, learning, and discussing with collaborators and colleagues. However, the researcher used this opportunity to study further the related concepts and methods (embedding problems in complexes and hypergraphs) in more detail and depth.
One of the main goals of this proposal was to contribute and extend the existing knowledge of expanders, by extending the theory of sparse robust expanders, and studying its relation to embedding problems. More specifically, one of the ambitious part of this proposal was to address a conjectures of Komlos from 1981, a major open problem in this field, both extremal and enumerative. In fact, Sharifzadeh and her collaborators proved a stronger version of this conjecture. The methods and tools developed in attacking this conjecture are likely to be useful for other embedding problems and edge-decomposition problems beyond the objectives in this proposal. For more details please see the technical report.
Also, during the fellowship, the researcher worked on some related problems, where she applied and further developed the techniques from Project 1.
In one project, the researcher and her collaborators studied the famous Erdős-Rademacher problem, which asks for the smallest number of copies of K_r (complete graph on r vertices) in a graph with the given number of vertices and edges. In early 2019, Sharifzadeh, together with her collaborators, described the asymptotic structure of all almost extremal graphs. To be more precise, let G_r (n,m) be the minimum number of copies of K_r in a graph with n vertices and m edges. they gave a description of up to o(n^2) edges of every graph with n vertices and m edges that has at most G_r(n,m)+o(n^r) many copies of K_r. Asymptotic structure results are often used in obtaining the exact value of the extremal function and the exact structure of the extremal graph. In fact, in almost all cases where the Erdős-Rademacher problem was solved exactly, the asymptotic structure result is used as a first step. In particular, knowing all extremal graphs within o(n^2) edges greatly helps in the ultimate aim for ruling out even a single adjacency that differs from a given set of extremal graphs. In other words, the results from this paper will be a possible first step for extremal graph theorists to solve the exact problem.
In another project, Sharifzadeh, together with her collaborators, work out a limit theory of Latin squares. Theories of limits of sequences of particular discrete structures have many applications. First and foremost, they have led to solutions of many asymptotic problems in extremal combinatorics. There are many representative samples for the class of graphs and also permutations. Sharifzadeh and her collaborators expect the theory that they develop in this project to have applications mostly in this area, that is, to answer asymptotic extremal questions about Latin squares. The set of tools provided in this project should be complete for such purposes.
Also, during the fellowship, the researcher worked on some related problems, where she applied and further developed the techniques from Project 1.
In one project, the researcher and her collaborators studied the famous Erdős-Rademacher problem, which asks for the smallest number of copies of K_r (complete graph on r vertices) in a graph with the given number of vertices and edges. In early 2019, Sharifzadeh, together with her collaborators, described the asymptotic structure of all almost extremal graphs. To be more precise, let G_r (n,m) be the minimum number of copies of K_r in a graph with n vertices and m edges. they gave a description of up to o(n^2) edges of every graph with n vertices and m edges that has at most G_r(n,m)+o(n^r) many copies of K_r. Asymptotic structure results are often used in obtaining the exact value of the extremal function and the exact structure of the extremal graph. In fact, in almost all cases where the Erdős-Rademacher problem was solved exactly, the asymptotic structure result is used as a first step. In particular, knowing all extremal graphs within o(n^2) edges greatly helps in the ultimate aim for ruling out even a single adjacency that differs from a given set of extremal graphs. In other words, the results from this paper will be a possible first step for extremal graph theorists to solve the exact problem.
In another project, Sharifzadeh, together with her collaborators, work out a limit theory of Latin squares. Theories of limits of sequences of particular discrete structures have many applications. First and foremost, they have led to solutions of many asymptotic problems in extremal combinatorics. There are many representative samples for the class of graphs and also permutations. Sharifzadeh and her collaborators expect the theory that they develop in this project to have applications mostly in this area, that is, to answer asymptotic extremal questions about Latin squares. The set of tools provided in this project should be complete for such purposes.