Fifteen points determine at least nine ordinary lines

An ordinary line of a finite point set in the real plane is a line containing exactly two of the points. Let m(n) be the minimum number of ordinary lines determined by n points, not all collinear. The values m(n) are known for n ≤ 14 and for the even n ≤ 24, where the Csima–Sawyer bound ⌈6n/13⌉ is attained by Böröczky's examples, while for n = 15 only 7 ≤ m(15) ≤ 9 has been known (the lower bound is the Kelly–Moser bound of 1958); the entry a(15) of OEIS sequence A003034 is the first unknown one. We prove that m(15) = 9, i.e. every set of 15 points, not all on a line, determines at least 9 ordinary lines; Böröczky's example (a regular octagon together with seven of the eight points at infinity of its sides and diagonals) shows that 9 is attained. Thus the Green–Tao formula (proved for all sufficiently large n) holds for all 6 ≤ n ≤ 16 except n = 13, and 15 points do not behave like McKee's sporadic 13-point set. The proof is a computer-assisted classification. Melchior's inequality and a counting argument reduce the question to the exact geometric realizability of the partial linear spaces on 15 points with one of seven prescribed line-size profiles (one with 7 and six with 8 ordinary lines). We enumerate these up to isomorphism (23,174,731 isomorphism classes in all) and show that none is realizable in the real projective plane with exactly the prescribed collinearities, using the Pappus and Desargues theorems as combinatorial filters and exact Gröbner-basis computations for the remaining 27,441 classes; all but one of the systems are rejected by arguments valid over every field of characteristic 0. The single remaining system, with three 4-point lines, one 5-point line and 23 three-point lines, has an exact realization over ℚ(√−3)—so 15 points of the complex projective plane can have 8 ordinary lines with a "real-looking" line profile—but it contains the Möbius–Kantor configuration and hence has no real realization. Every algebraic rejection is confirmed by an independent SAT-based test for the existence of a rank-3 oriented matroid, and the same pipeline recovers the known extremal configurations for 12, 13 and 14 points (and shows that they are the only ones), which serves as a positive control. The authors are listed in alphabetical order and contributed equally.