Are there arbitrarily large sets $S$ of natural integers such that the difference of each pair is their GCD? I am interested in sets $S \subseteq \mathbb{N}$ of natural integers with the following property: for any $i, j \in S$, then the greatest common divisor of $i$ and $j$ is the absolute value of their difference, $|i - j|$. Equivalently, this amounts to requiring that the difference $|i-j|$ divides $i$ and $j$.
Are there sets $S$ of arbitrarily large cardinality with this property?
Examples of such sets $S$ of cardinality up to 9 are given in sequence A213918 of the OEIS, but there is no discussion there about whether arbitrarily large such sets exist.
My intended use for this would be to obtain a variant of Van der Waerden's theorem, to show that from an infinite word one can extract arbitrarily large monochromatic arithmetic progressions with the added condition that the common difference of the arithmetic progression divides its first term.
 A: DEFINITION: Positive integers $a$ and $b$ are Special if $\gcd(a,b) = |a-b|$.
LEMMA$1$: Positive integers $a$ and $b$ are Special if and only if $a \equiv 0 \bmod{|b-a|}$.
PROOF: Suppose that $a$ and $b$ are Special. Then $|b-a|$ divides $a$. Hence $a \equiv 0 \bmod |a-b|$.
Suppose that $a \equiv 0 \bmod |b-a|$. If $a > b$, then $b = a - |b-a|$. If  $a < b$, then $b = a + |b-a|$. In either case $b \equiv 0 \bmod |b-a|$. Since 
$b-a = \pm |b-a|$ and $|b-a|$ divides both $a$ and $b$, it follows that $\gcd(a,b) = |b-a|$.
DEFINITION: We say that an increasing sequence of positive integers 
$\mathbf s = \{s_0,\; s_1,\; s_2,\; \dots,\; s_n\}$ is Special if, for every $0 \le i < j \le n$, $s_i$ and $s_j$ are special.
An immediate consequence of LEMMA$1$ is
THEOREM$1$: Suppose that the increasing sequence
$\mathbf s = \{s_0,\; s_1,\; s_2,\; \dots,\; s_n\}$ 
is Special and let 
$\sigma = \operatorname{lcm}\left\{ s_i \right\}_{i=0}^n$.
then the sequence
$\mathbf s'=\{\sigma,\; \sigma+s_0,\; \sigma+s_1,\; \sigma+s_2,\; \dots,\; \sigma+s_n\}$ 
is Special
For example, the sequence $\mathbf s = \{2,3,4\}$ is Special and
$\sigma = \operatorname{lcm}\{ 2,3,4 \} = 12$. Then 
 $\mathbf s' = \{12,14,15,16\}$ is Special.
$\mathbf s''=\{1680, 1692, 1694, 1695, 1696\}$.
ADDENDUM (4/5/2017)
We will present an example from which the more general case should be clear.
Suppose $A < B < C < D < E$ and we wish to test whether or not 
$\mathbf S = \{A,B,C,D,E\}$ is Special.
This will be equivalent to showing that the following pairs are special:
$$\begin{array}{cccc}
(A,B) &(A,C) &(A,D) &(A,E) \\
      &(B,C) &(B,D) &(B,E) \\
            &&(C,D) &(C,E) \\
                  &&&(D,E) \\
\end{array}$$
Which is equivalent to showing that
$$\begin{array}{cccc}
B-A \mid A & C-A \mid A & D-A \mid A & E-A \mid A \\
           & C-B \mid B & D-B \mid B & E-B \mid B \\
                       && D-C \mid C & E-C \mid C \\
                                   &&& E-D \mid D
\end{array}$$
We "encode" this as
$$\begin{array}{lrrrr}