Construct a set of points having some properties (colouring related) 
Construct a finite set of points $S$, all in the same plane, such
  that:
  
  
*
  
*Every line in the plane intersects $S$ in no more than $4$ points.
  
*If the points of $S$ are arbitrarily coloured with two colours, there are 3 collinear points of $S$ having same colour.
  


I think the following set $S$ of $13$ points would solve the problem (the horizontal lines are parallel):

Obviously the first condition is fulfilled.
Now, suppose there are two colours, blue and green. It seems that no matter how I distribute them, I end up with having 3 points with the same colour on the same line.
But I cannot give a rigorous proof and I need help with this (of course, listing all the possibilities is not an option).

Suppose there is a colouring scheme that contradicts the second requirement. Then on every 4-points-line one must have 2 green points and 2 blue points. 
Let's set a weight for each point: if the colour is blue the weight is 1 otherwise the weight is -1.
Now, let's calculate the weight's sum for all points in two ways:
1) Add all the weights on the same horizontal line. Then $$Sum = 1 + 0 + 0+0$$
2)  Add all the weights on the oblique lines (I'll consider the upper blue point on the first oblique line from the left). Then 
$$ Sum = 0 + (-1) + (-1) + (-1) = -3$$
The two different values for the sum provides the contradiction.
