Abstract

Sum-free subsets

Ian Levitt
Wed Nov 10 at 11:30am in the Graduate Student Lounge

A subset $A$ of an Abelian group is called sum-free if $(A+A) \cap A = \emptyset$. In 1965, Paul Erd\"os proved that any subset $B$ of non-zero integers contains a sum-free subset of size at least $|B|/3$. Alon and Kleitman's similar argument in 1990 shows that this bound is in fact strict. They go on to show that the constant $1/3$ cannot be replaced by $12/29$, and that for general Abelian groups, $2/7$ is the best constant. In 1997, Jean Bourgain formulates Erd\"os' proof into Harmonic Analysis and manages to show that any $B \subset \Z-{0}$ contains a sum-free subset of size at least $(|B| + 2)/3$. We will discuss as much of this as time allows.