Abstract
An ordinary circle of a set P of n points in the plane is defined as a circle that contains exactly three points of P. We show that if P is not contained in a line or a circle, then P spans at least n^2/4 - O(n) ordinary circles. Moreover, we determine the exact minimum number of ordinary circles for all sufficiently large n and describe all point sets that come close to this minimum. We also consider the circle variant of the orchard problem. We prove that P spans at most n^3/24 - O(n^2) circles passing through exactly four points of P. Here we determine the exact maximum and the extremal configurations for all sufficiently large n. These results are based on the following structure theorem. If n is sufficiently large depending on K, and P is a set of n points spanning at most Kn^2 ordinary circles, then all but O(K) points of P lie on an algebraic curve of degree at most four. Our proofs rely on a recent result of Green and Tao on ordinary lines, combined with circular inversion and some classical results regarding algebraic curves.
Highlights
1.1 BackgroundThe classical Sylvester–Gallai theorem states that any finite non-collinear point set inR2 spans at least one ordinary line
More sophisticated statement is the so-called Dirac–Motzkin conjecture, according to which every non-collinear set of n > 13 points in R2 determines at least n/2 ordinary lines. This conjecture was proved by Green and Tao [13] for all sufficiently large n. Their proof was based on a structure theorem, which roughly states that any point set with a linear number of ordinary lines must lie mostly on a cubic curve
Our first main result concerns the minimum number of ordinary circles spanned by a set of n points, not all lying on a line or a circle, and the structure of sets of points that come close to the minimum
Summary
The classical Sylvester–Gallai theorem states that any finite non-collinear point set in. A more sophisticated statement is the so-called Dirac–Motzkin conjecture, according to which every non-collinear set of n > 13 points in R2 determines at least n/2 ordinary lines This conjecture was proved by Green and Tao [13] for all sufficiently large n. Using group laws on certain cubic curves, one can construct n non-collinear points with n(n − 3)/6 + 1 3-point lines, and Green and Tao [13] proved (for large n) that this is optimal This does not follow directly from the Dirac–Motzkin conjecture, but it does follow from the above-mentioned structure theorem of Green and Tao for sets with few ordinary lines (Theorem 5.1). We do not consider other related problems, we remark that similar questions have been asked for ordinary conics [7,10,25], ordinary planes [2], and ordinary hyperplanes [3]
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