Green 2013 sets defining few ordinary lines

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proposition_2_1: For each integer m >= 3, the set X_{2m} has 2m points and m ordinary lines, and three modifications of X_{4m} and X_{4m+2} give 4m+1 points with 3m ordinary lines and 4m-1 points with 3m-3, so f(n) ordinary lines are attained for every such n.

proposition_2_6: A subgroup of order n >= 3 of the nonsingular points of an irreducible cubic curve spans n - 1 - 2·1_{3|n} ordinary lines and floor(n(n-3)/6)+1 lines through exactly three of its points.

theorem_1_2: Green and Tao's proof of the Dirac-Motzkin conjecture for large n: a finite set of n points in the plane, not all on one line, spans at least n/2 ordinary lines once n is at least an absolute constant n_0.

theorem_1_3: Green and Tao's solution of the orchard problem for large n: a finite set of n points in the plane, n at least an absolute constant n_0, has at most floor(n(n-3)/6)+1 lines containing exactly three of its points.

theorem_1_4: If n points in the plane span at most Kn ordinary lines with K >= 1 and n >= exp exp(CK^C) for a large absolute constant C, then all but at most O(K^{O(1)}) of the points lie on an algebraic curve of degree at most 3.

theorem_1_5: If n points of the real projective plane span at most Kn ordinary lines and n >= exp exp(CK^C), then after a projective transformation they differ in O(K) points from n - O(K) collinear points, from the set X_{2m} with m = n/2 + O(K), or from a coset of a finite subgroup of an irreducible cubic.

theorem_2_2: For n at least some n_0, n points of the real projective plane not all on a line span at least f(n) ordinary lines, where f(2m) = m, f(4m+1) = 3m and f(4m-1) = 3m-3, and equality holds only for the Böröczky examples up to a projective transformation.

theorem_2_4: There is an absolute constant C such that any n points of the real projective plane, not all on a line, spanning at most n - C ordinary lines are projectively equivalent to a Böröczky example or a near-Böröczky example.

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Ben Green, Terence Tao, On sets defining few ordinary lines. Discrete & Computational Geometry 50 (2013), no. 2, 409-468. arXiv:1208.4714, doi:10.1007/s00454-013-9518-9. The copy read for this card is arXiv:1208.4714v3 (28 March 2013), and the labels below are that version's. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1208.4714), every other right reserved.

The paper proves that a set P of n points in the plane, not all collinear, spans at least n/2 ordinary lines (lines with exactly two points of P) once n is large (Theorem 1.2, the Dirac-Motzkin conjecture), improving on the 3n/7 of Kelly-Moser and the 6n/13 (for n > 7) of Csima-Sawyer; Theorem 2.2 gives the exact minimum for large n, f(n) with f(2m) = m, f(4m+1) = 3m and f(4m-1) = 3m-3 (so 3*floor(n/4) when n is odd), attained only by the Böröczky examples of Proposition 2.1 up to projective transformation. Theorem 2.4 shows that every set with at most n - C ordinary lines, C an absolute constant, is a Böröczky or near-Böröczky example. Theorem 1.3 solves the orchard problem for large n: at most floor(n(n-3)/6)+1 lines are 3-rich, attained by the subgroups of cubic curves of Proposition 2.6. Underlying both are structure theorems for sets with at most Kn ordinary lines and n large in terms of K: all but O(K^{O(1)}) points lie on a curve of degree at most 3 (Theorem 1.4), and, more precisely, the set is within O(K) points of a line, of the set X_{2m}, or of a coset of a finite subgroup of an irreducible cubic (Theorem 1.5, proved in Section 7). The paper proves Theorem 2.4 by applying Theorem 1.5, noting that its polynomial-error form, Theorem 7.1, suffices there, and proves Theorem 1.3 from Theorem 1.5, remarking that the intermediate structure theorem, Proposition 5.3, would suffice for the latter. The method combines Melchior's projective-dual proof of Sylvester-Gallai with Euler's formula, group laws on cubic curves, and additive-combinatorial tools (Appendix A) plus intersections of lines through roots of unity (Appendix B).

Source: https://arxiv.org/abs/1208.4714.

Bears on.

• #210: Theorem 2.2 gives the least number of ordinary lines spanned by n points not all on a line as n/2 for even n and 3*floor(n/4) for odd n, for all n >= n_0 with n_0 not made explicit; Theorem 1.2 is its lower bound n/2 and Proposition 2.1 gives the matching examples.
• #669: Theorem 1.3 with Proposition 2.6 gives f_3(n) = floor(n(n-3)/6)+1, lines through exactly 3 points, for all large n. The paper does not treat F_k(n), lines through at least k points, or any k >= 4.

Read status. Claims checked: the statements on the result pages were read clause by clause on the print; no proof was checked. The second sentence of Proposition 2.6, on shifted cosets, is inconsistent as printed with the pair-counting identity the proof uses, and the corpus does not use it (see that page).

Results. Labels and pages are those of arXiv:1208.4714v3.

• Theorem 1.2 (p. 2): for n >= n_0, n points in the plane not all on a line span at least n/2 ordinary lines.
• Theorem 1.3 (p. 3): for n >= n_0, n points in the plane have at most floor(n(n-3)/6)+1 lines through exactly three of them.
• Theorem 1.4 (p. 5): with at most Kn ordinary lines, K >= 1 and n >= exp exp(CK^C), all but O(K^{O(1)}) points lie on a curve of degree at most 3.
• Theorem 1.5 (p. 6): with at most Kn ordinary lines, K > 0 and n >= exp exp(CK^C), after a projective transformation P differs in O(K) points from n - O(K) points on a line, from X_{2m} with m = n/2 + O(K), or from a coset H + g, 3g in H, of a finite subgroup H of size n + O(K) of the nonsingular real points of an irreducible cubic; each such set has O(Kn) ordinary lines.
• Proposition 2.1 (p. 10): for m >= 3, X_{2m} has m ordinary lines, X_{4m} plus the origin and X_{4m+2} minus a point at infinity have 4m+1 points and 3m, and X_{4m} minus [0,1,0] has 4m-1 points and 3m-3.
• Theorem 2.2 (p. 13): for n >= n_0, n points of RP^2 not all on a line span at least f(n) ordinary lines, with equality only for the Böröczky examples up to projective transformation.
• Theorem 2.4 (p. 14): for an absolute constant C, a set not all on a line with at most n - C ordinary lines is projectively a Böröczky or near-Böröczky example.
• Proposition 2.6 (pp. 18-19): a subgroup of order n >= 3 of the nonsingular points of an irreducible cubic spans n - 1 - 2*1_{3|n} ordinary lines and floor(n(n-3)/6)+1 3-rich lines.

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