There are $K$ points in the space, no three of which are colinear. All pairs of points are connected by line segments and each segment is colored either yellow or green such that the following two conditions are held:
- There is no triangle with exactly 1 yellow side .
- There are no 4 points any two connected by the same colored segment. What is the maximal possible value of $K$?
Condition 1 implies that the green segments define an equivalence relation (if you also consider every point to be equivalent to itself).
Applying condition 2 to green means that no equivalence class can have more than three points. Applying condition 2 to yellow means there can't be more than three equivalence classes. So $K\leq 9$; it should now be easy to construct an example with $K=9$.