Find the greatest positive integer $n$ such that we can choose $2007$ different positive integers from $[2\cdot 10^{n-1},10^{n})$ such that for each two $1\leq i<j\leq n$ there exists a positive integer $\overline{a_{1}a_{2}\ldots a_{n}}$ from the chosen integers for which $a_{j}\geq a_{i}+2$.
A. Ivanov, E. Kolev
$ 1)$ Notice that for all the $ 2007$ numbers $ a_1 \ge 2$
$ 2)$ If $ n=4k$ then a choice covering the maximum amount of pairs is $ 24682468\cdots2468$ covering $ 6 \cdot \frac{k(k+1)}{2}$ pairs
$ 3)$ There are $ \frac{n(n+1)}{2}$ pairs
mszew wrote:
$ 2)$ If $ n = 4k$ then a choice covering the maximum amount of pairs is $ 24682468\cdots2468$ covering $ 6 \cdot \frac {k(k + 1)}{2}$ pairs
Umm... how can you prove this is maximum?
Moreover, the other $ n$s won't cover this many pairs...