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.
