Some interesting problems

There are some interesting problem, I post them at there in case I forget them. Excuse me if they are trivial, I have not took enough time to consider them about I think they are valuable to be consider.

Problem 1:

This problem is stated by graph coloring. there are two prat of it, in fact the first part I heard from someone else and I try to generate it to high dimension.

  1. there are finite lines $latex \{l_i\}_{i\in I}, l_i\subset \mathbb R^2$, crossing each other and the is a set $latex J$ of crossing point. for technique reason, assume the position of lines are generic, i.e. no three of them intersect at one point. Then we could use 3 different colors to color  $latex J$ make Neighbor points have different color. And to proof 3 is smallest.
  2. generate it to high dimension, to prove $latex \mathbb R^n$ case, $latex n+1$ is the number.

This seems to be a graph problem, but the underlying structure is linear structure and some topological obstacle. I am not very sure. But it seems we can use an energy decrement argument with the obesevation:

The existence of a reasonable definition of “energy of correlation”.

the simplex arrive with the maximum of “correlation energy” in a very symmetric way, and this situation is easy to handle (coloring).

If make sense, this argument could also generate to high dimension.

Problem 2:

Let us consider some example of map between two metric space, a toy model is a line and two parallel lines, I called two parallel lines by $latex X_1\cup X_2$, the single line by $latex X_3$. The problem is try to find a tuple $latex (d,f)$, where $latex d$ is a metric define on $latex X_1\cup X_2$ and $latex f: X_1\cup X_2\to X_3$. such that the distortion of $latex f^* d$ and the standard metric on $latex X_3$ arrive at a infimum, this of course could not be the case, such like the situation of Yamabe problem on manifold with conners. So, let us ask a more general problem, could we describe the behavior of $latex f$ in some sense? what could we say with this kind of $latex f$?