SMF

On the structure and the number of sum-free sets

On the structure and the number of sum-free sets

Gregory A. FREIMAN
  • Année : 1992
  • Tome : 209
  • Format : Papier
  • Langue de l'ouvrage :
    Anglais
  • Class. Math. : 11B75
  • Pages : 195-201
  • DOI : 10.24033/ast.163

A finite set $A$ of positive integers is called sum-free if $A\cap (A + A) = \emptyset $. For $n$ odd, $\{1, 3, 5, \ldots , n\}$ and $\{\frac {n+1}2,\frac {n+3}2,\ldots , n\}$ are examples of such sets. Denote by $m$ and $l$, respectively, the largest and smallest elements of $A$ and by $a$ the cardinality of $A$. We show that if the cardinality of the sum-free set $A$ does not differ much from $\frac l2$, then $A$ does not differ much from one of the two examples mentioned above. More precisely, if $a >\frac 5{12}l + 2$, then either all elements of $A$ are odd or $A$ contains both odd and even integers and $m\geq a$. It is shown that if $a > \frac 5{12}l + 2$ then the number of such sum-free sets is $O(2^{n/2})$ which proves for such sets the conjecture of P. Cameron and P. Erdös.

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