The following question was asked on math.stackexchange, where it received no answers.

https://math.stackexchange.com/questions/1392669/maximum-size-of-a-union-of-incomparable-chains

Let $\mathbb{N}^{<\mathbb{N}}$ denote the set of finite sequences of natural numbers (not including the empty sequence). Order this set by $E\preceq F$ if $E$ is an initial segment of $F$. Call a collection $(s_i)_{i=1}^n$ of subsets of $\mathbb{N}^{<\mathbb{N}}$ incomparable if for $1⩽i,j⩽n$ with $i≠j$, no member of $s_i$ is comparable to any member of $s_j$. Given a finite subset $A$ of $\mathbb{N}^{<\mathbb{N}}$, what is a lower estimate on the largest cardinality of a union of incomparable chains in $A$, as a function of the cardinality of $A$? In particular, is there a number $c>0$ so that any finite subset $A$ admits a subset $B$ which is a union of incomparable chains so that $|B|⩾c|A|$?

membersof $s_i$ are incomparable to themembersof $s_j$ when $i$ and $j$ are distinct. $\endgroup$ – user78375 Aug 20 '15 at 18:00