Let $\sigma :\mathbb{N}\rightarrow\mathbb{R}$ an injective sequence of real numbers.


There exists an infinite set $A=$ { ${a\_{1},a\_2,\ldots ,a\_n,\}\ldots$ } $ \subset{N}$ such that 


i) $\sigma\_{|A}$ is monotone


ii) $a\_n=O(n^2)$ ?



Chains or Antichains slowly increasing
For $0 \leq k \lt 2^j$ , let $\sigma(k+2^j)=(2k+1)/2^{j+1}$ . Let A be a subset of integers such
that $\sigma\mid\_A$ is monotonic. Then $a\_{n+1} - a\_n$ is greater than $a\_n/4$ infinitely often,
which cannot hold if $a\_n$ is $O(n^d)$ for any positive integer $d$.


Gerhard "Can't Make It Much Simpler" Paseman, 2013.04.22 


