While reading about Sylvester–Gallai theorem, I got the simple proof of the theorem using Induction.
Statement:
Given a finite set of points in Euclidian plane such that any line passing through two of the points shall pass through at least one more of them, then all points have to be collinear.
Base case:
Let n=3
Given 3 points, If a line containing two of them passes through one more of them, they are by definition, collinear.
Thus statement #1 is true for n=3.
Step Case
P(k) : Given k points. Assuming that the statement #1 holds true for given set of k points.
P(k+1):
Show that for every k ≥ 3, if P(k) holds, then P(k + 1) also holds.
Here we can say that for a new point (k+1 th) in the same plane to pass through any line passing through two of the points in the k-points set in P(k), then only the newly added point can go through atleast 3 points. If it pass through atleast two of the point then it has to pass through the line passing through all of them.
Hence the only way to expand the given set of k points to a set of k+1 number of points, is to include an external point which is collinear with the existing points.
Hence all k+1 points of the expanded set are collinear.
Thus by principle of induction, (1) is true for all n>=3.
Is this proof complete ?
Best Answer
The issue here is that you can't apply the induction hypothesis.
Although you know that the set of $k+1$ points satisfy the conditions, that doesn't mean that you can just throw out 1 point and make the remaining $k$ points still satisfy the condition, as you potentially need that thrown out point to serve as the "another point on the line for some $p_i, p_j$".
Explicit example from the OP (See comments)