Extremal Product-one free sequences in some non-abelian groups and shifted turan sieve method on tournaments

Abstract

This thesis is divided into two parts: Part I: Inverse zero-sum problems.We start presenting an overview on Zero-Sum Theory. In particular, we present the main results and conjectures concerning the following invariants: Davenport constant, Erd}os-Ginzburg-Ziv constant and n constant (either with weights f1g or unweighted as well). Afterwards, we focus on Davenport constant: this invariant denotes the smallest positive integer D(G) such that every sequence of elements in G of length jSj D(G) contains a product-1 subsequence in some order, where Gis a finite group written multiplicatively. J. Bass [6] determined the Davenport constant of metacyclic (in some special cases) and dicyclic groups, and J. J. Zhuang and W. D. Gao [88] determined the Davenportconstant of dihedral groups. In a joint work with F. E. Brochero Martínez (see [11] and [12]), for each of these non-abelian groups we exhibit all sequences of maximum length that are free of product-1 subsequences.We conclude by presenting Properties B, C, and D, which are, in general, conjectures of extreme importance in the study of inverse problems. Part II: Shifted Turán sieve on tournaments. We start presenting an overview on some famous problems which can be partially or totally resolved using Sieve Theory. Afterwards, we construct a shifted version of the Turán sieve method, developed by Y.-R. Liu and M. R. Murty (see [54] and [55]), and apply it to counting problems on tournaments in graph theory, i.e., complete directed graph, according to the number of cycles. More precisely, we obtain upper bounds for the number of tournaments which contain a small number of restricted r-cycles (in case of normal or multipartite tournaments) or unrestricted r-cycles (in case of bipartite tournaments), as done in [53]. Then, we show how the sieve theory may in the future be useful in other mathematical branches, such as zero-sumproblems.