Equal coloring of balanced ternary tree 
Given a complete balanced ternary tree (every leaf is at the same depth) where every edge has length $1$. An equal coloring is a coloring of the edges in red and black (each edge can be broken up into several intervals of alternating colors) such that the length of the red part is equal to the length of the black part.
Fix a positive integer $n$. Does there exist $h$ such that if the tree has height $h$, then in any equal coloring, the part corresponding to some color has more than $n$ connected components?

For $n=1$, we may take $h=1$ (so the tree has only three edges). It is easy to see that any coloring such that each color has length $1.5$ must entail some color being broken into two connected components. With $n=2$, taking $h=1$ or $h=2$ is not enough.
 A: I misinterpreted the question to mean that the vertices are also colored, so a vertex cannot be incident to two connected red edges while also being incident to two connected black edges. The correct interpretation is that only the interior of the edges is colored - the vertices are uncolored (or perhaps multicolored is a good way to think of it). So a set $S$ of red points is connected if $S$ is in the same connected component of the tree when you delete the black part.
Fortunately we can reduce the original problem to the problem I was thinking of. Color each non-root vertex according to the nearest color used on the edge leading to the root vertex, and color the root arbitrarily. Each connected component is split into at most three components by this process. So if the original tree had $n'$ connected components, the new tree has at most $n=3n'$ connected components. Use this $n$ in the following argument.

Yes, we can use any $h$ with $3^h>2(n+1)3^n.$
Let $R_1,\dots,R_r$ be the connected components of the red part, and $B_1,\dots,B_b$ the connected components of the blue part.
For any set $S$ let $S^*$ denote the set of all points $p$ such that the shortest path from $p$ to the root of the tree goes through $S.$ (Sort of like taking all descendents, but including points in edges as well.)
If the root of the tree is red, then the total length of the red part is
$$L(R_1\cup\dots\cup R_r)=L(R^*_1)+\dots+L(R^*_r)-L(B^*_1)-\dots-L(B^*_b)$$
where $L$ means total length. This is a form of the exclusion-inclusion formula.
For any connected set $S$ there is a unique point $p$ in the closure of $S$ nearest to the root. The distance of $p$ to a leaf can be written as $d_0+D$ where $D$ is an integer and $d_0\in[0,1)$ if $p\in S,$ and $d_0\in(0,1]$ if $p\not\in S.$ Then
$$L(S^*)=d_0+3+3^2+\dots+3^D=d_0+\tfrac32(3^D-1).$$
For example if $S^*$ consists of a vertex, three child leaves and the edges in between, then $D=1$ and $L(S^*)=3.$
The total length of the tree is $\tfrac32(3^h-1).$ So it suffices to show that a sum of $n$ numbers of the form $\pm (d_0+\tfrac 32(3^D-1))$ cannot equal $\tfrac34(3^h-1).$
Dividing through by $\tfrac32 3^h,$ we want to show that a sum of $n$ numbers of the form $\pm 3^{D-h}$ cannot equal $\tfrac 12$ to within additive error $3^{-h}(n+1).$
Using $3^h>2(n+1)3^n$ this follows from:

For integer $n\geq 1$ and integers $x_1,\dots,x_n,$ we have $$|\pm 3^{x_1}\pm\dots\pm 3^{x_n}-\tfrac12|\geq \tfrac12 3^{-n}.$$

Using the rules $3^x+3^x=3^{x+1}-3^x$ we can reduce to the case where the $x_i$ are distinct - a balanced ternary representation. Wlog $x_1$ is the largest. The sum $3^{x_1}\pm\dots\pm 3^{x_n}$ lies in the range $(\tfrac12 3^{x_1},\tfrac 32 3^{x_1}),$ which means we're done unless $x_1\in\{0,-1\}$ - we don't need to consider cases where $3^{x_1}\pm\dots\pm 3^{x_n}$ is less than $\tfrac16$ or greater than $\tfrac32.$
The case $n=1$ is just arithmetic: $|1-\tfrac12|,|\tfrac13-\tfrac12|\geq \tfrac16.$
For $n>1$ we can use induction; the quantity $t=\pm 3^{x_2}\dots\pm 3^{x_n}$ satisfies $|-t-\tfrac 12|,|t-\tfrac 12|\geq \tfrac12 3^{-(n-1)}.$ For $x_1=0$ use $$|1+t-\tfrac12|=|t+\tfrac12|=|-t-\tfrac12|\geq\tfrac12 3^{-(n-1)}\geq\tfrac12 3^{-n}.$$
For $x_1=-1$ use
$$|\tfrac13+t-\tfrac12|=\tfrac13|3t-\tfrac12|\geq \tfrac12 3^{-n}.$$
