A friend of mine came up with the following problem:
Let $\{X_1, X_2, ..., X_n\}$ be an arbitrarily finite partition of the unit square $[0, 1]^2$. Let $\{P_1, P_2, ..., P_m\}$ be a finite set of points in $[0, 1]^2$.
Can all the points $P_i$ be transformed through the same bijective affine transformation (rotation, translation, squeezing, scaling and shearing), so that all $P_i$ are contained in one of the $X_k$ ?
On a side note: Our professor told us to maybe ask somebody from the graph theory department. Could someone explain what the connection to graph theory is?
Edit: I would also be interested in ideas about the one-dimensional analogue.