The length of an <i>s</i>-increasing sequence of <i>r</i>-tuples

We prove a number of results related to problem Po-Shen Loh, which is equivalent in Ramsey theory. Let $a=(a_1,a_2,a_3)$ and $b=(b_1,b_2,b_3)$ be two triples integers. Define $a$ 2-less than $b$ if $a_i<b_i$ for at least values $i$, define sequence $a^1,\dots,a^m$ 2-increasing $a^r$ $a^s$ whenever $r<s$. Loh asks how long can all the take $\{1,2,\dots,n\}$, gives $\log_*$ improvement over trivial upper bound $n^2$ by using triangle removal lemma. In other direction, simple construction lower $n^{3/2}$. look this collection generalizations, improving some known bounds, pointing out connections well problems extremal combinatorics, asking further questions.