(Remark. A set that intersects every line of the plane in exactly n points is called an n-set. A 1-set is clearly impossible; this question asks whether 2-sets exist.)

Hint 1

The answer is Yes, although it's hard to visualize such a set. This result was proved by Stefan Mazurkiewicz in 1914, and we will show his proof. In this article we will show how to construct the set using transfinite recursion. Surprisingly the construction uses almost no facts about geometry or the plane, but it does use some facts about transfinite numbers, so you should be familiar with those. (Most books on set theory discuss this topic.)

Let's practice on a simpler version of the problem before we take a leap into the transfinite. Suppose we are given a finite collection of straight lines in the plane, L₁, ..., L_N. Describe a method for generating a set of points in the plane such that (1) each line of this collection intersects exactly two members of the set, and (2) any line in the plane intersects at most two members of the set. An easy way to do this would be to draw a very large circle, large enough that every line intersects it; but this method does not generalize well, so think of some other method where you select the points one at a time.

Hint 2

The set of points can be defined in a simple way by induction. We will define the sequence of points y₁, y₂, .... We'll also define the sets A₂, A₃, ... by A_k = {y₁, y₂, ..., y_k-1}. We write A for the set of all y_k.

Define y₁ as any point of L₁.
Having defined y₁, ..., y_k-1, define y_k as follows. If all the L_i intersect A_k in at least two points, we are finished. Otherwise, pick the smallest i such that L_i intersects A_k in fewer than two points. Choose y_k as any point of L_i that is not already in A_k and that is not co-linear with any two points of A_k.

By construction, no three points of any A_k are co-linear, so no line in the plane intersects A in more than two points. The construction clearly terminates after at most 2N steps, because each step adds a point and there are at most 2 points per line.

Note that this construction gradually works its way up through the lines. In general we add two points on each line, then step to the next-numbered line, and in the general case we end up with a set containing 2N points. It's possible that the final set may be smaller; it may happen that an early point also occurs in a later line, so when we get to the later line we don't have to add two points.

Now we have to make the leap to transfinite numbers. Our constructions works well for a finite collection of lines; how can we extend it to the collection of all lines of the plane?

Click here for the complete solution.