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   Seminário de Teoria da Computação e Combinatória (TCC)

             Minioficina de Combinatória
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Título:      Counting sum-free sets in Abelian groups
            
Palestrante: Rob Morris
             Instituto de Matemática Pura e Aplicada

Hora e Data: 15h10m, sexta-feira, 09 de setembro de 2011

Local:       auditório do NUMEC

Resumo: 

Sum-free sets, i.e., sets containing no solution to 
           
      x + y = z, 

play an  important role in Additive Number  Theory, and much
progress  has been  made on  understanding  their properties
over the past ten years or so.  Perhaps the most famous such
result is  the Cameron-Erdos Conjecture, proved  by Green in
2004,  which  states  that  there  are  O(2^{n/2})  sum-free
subsets of  [n].  For finite  Abelian groups G of  "Type I",
Green and Ruzsa proved  even sharper bounds, determining the
asymptotic number of sum-free sets in G.

In this talk we shall show how to transfer the theorems of
Green and Green-Ruzsa to the sparse setting; in particular,
we show that there are 

      2^{O(n/m)} {n/2 \choose m} 

sum-free subsets of [n] of size m, for every
 
      m \ge C \sqrt{n \log n}. 

Our main tool is  a new transference principle for 3-uniform
hypergraphs, which has several other applications.