Master's Dissertation
DOI
https://doi.org/10.11606/D.45.1996.tde-07052010-163719
Document
Author
Full name
Cesar Alberto Bravo Pariente
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 1996
Supervisor
Committee
Kohayakawa, Yoshiharu (President)
Grable, David Alan
Title in Portuguese
Um método probabilístico em combinatória
Keywords in Portuguese
Combinatória
Geometria
Método Probabilístico
Teoria de Números
Abstract in Portuguese
Title in English
A Probabilistic Method in Combinatorics
Keywords in English
Combinatorics
Geometry
Number Theory
Probabilistic Method
Abstract in English
The following work is an effort to present, in survey form, a collection of results that illustrate the application of a certain probabilistic method in combinatorics. We do not present new results in the area; however, we do believe that the systematic presentation of these results can help those who use probabilistic methods comprenhend this useful technique. The results we refer to have appeared over the last decade in the research literature and were used in the investigation of problems which have resisted other, more classical, approaches. Instead of theorizing about the method, we adopted the strategy of presenting three problems, using them as practical examples of the application of the method in question. Surpisingly, despite the difficulty of solutions to these problems, they share the characteristic of being able to be formulated very intuitively, as we will see in Chapter One. We should warn the reader that despite the fact that the problems which drive our discussion belong to such different fields as number theory, geometry and combinatorics, our goal is to place emphasis on what their solutions have in common and not on the subsequent implications that these problems have in their respective fields. Occasionally, we will comment on other potential applications of the tools utilized to solve these problems. The problems which we are discussing can be characterized by the decades-long wait for their solution: the first, from number theory, arose from the research in Fourier series conducted by Sidon at the beginning of the century and was proposed by him to Erdös in 1932. Since 1950, there have been diverse advances in the understanding of this problem, but the result we talk of comes from 1981. The second problem, from geometry, is a conjecture formulated in 1951 by Heilbronn and finally refuted in 1982. The last problem, from combinatorics, is a conjecture formulated by Erdös and Hanani in 1963 that was treated in several particular cases but was only solved in its entirety in 1985.