Discoveries / number-theory

Erdos-Graham Question on Averages of Unit Fractions

Erdos and Graham asked whether a positive-density subset of $\{1,\ldots,N\}$ can avoid having any two distinct elements $a,b$ whose unit fractions average to a unit fraction. It can: there is a constant $c>0$ such that for all large $N$ some $A \subseteq \{1,\ldots,N\}$ of size $> cN$ has that property, which also gives the best known lower bounds for related unit-fraction avoidance problems.