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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.

Let a finite set of ≥ d + 1 points in the plane with pairwise distinct x coordinates be given. Assume that any graph of a degree ≤ d polynomial through any d + 1 of the points passes through at least one more of the given points. Prove that all the given points belong to the graph of a polynomial of degree ≤ d.

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    $\begingroup$ Hint: if f(x) is a polynomial of degree at most d, and x_0 is a real number, then (f(x)-f(x_0))/(x-x_0) is a polynomial of degree at most d-1. Use this and induction on d. $\endgroup$
    – Terry Tao
    Commented May 21, 2017 at 17:28
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    $\begingroup$ See also Boys et al., On the number of ordinary conics, SIAM J Discrete Math 30 (2016) 1644-1659, MR3539894. $\endgroup$
    – Gerry Myerson
    Commented May 22, 2017 at 3:43

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