According to the Sylvester–Gallai theorem, given a finite number of points in the Euclidean plane, either: 1) all the points are collinear; or \ 2)there is a line which contains exactly two of the points. \ Now, I want to know, is it possible to generalize these theorem as follows.
Given a finite number of points in the Euclidean plane and $k\geq 2$, either \ 1) all of them lies on the polynomial of the form $a_kx^k+\cdots +a_1x+a_0$; or \ 2) there exist a polynomial of the form $P(x)= a_kx^k+\cdots +a_1x+a_0$ which contains exactly $k+1$ of them.