Triple-Free Set

A Set of Positive integers is called weakly triple-free if, for any integer $x$, the Set $\{x, 2x, 3x\}\not\subset S$. It is called strongly triple-free if $x\in S$ Implies $2x\not\in S$ and $3x\not\in S$. Define

$$\begin{eqnarray*} p(n)&=&\max\{\vert S\vert: S\subset \{1, 2, \ldots, n\} \hbox{... ...set \{1, 2, \ldots, n\} \hbox{\rm\ is\ strongly\ triple-free}\}, \end{eqnarray*}$$

where $\vert S\vert$ denotes the Cardinality of $S$, then

$$\lim_{n\to\infty} {p(n)\over n}\geq {\textstyle{4\over 5}}$$

and

$$\lim_{n\to\infty} {q(n)\over n}=0.6134752692\ldots$$

(Finch).

See also Double-Free Set

References

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/triple/triple.html