The orchard-planting problem for fifteen points

Let t₃(n) be the maximum number of lines containing exactly three points of a set of n points in the real plane. The values t₃(n) are known for n ≤ 14 and for n = 16, and it has been known since 1974 that t₃(15) ∈ {31, 32}. We prove that t₃(15) = 31: fifteen points determine at most 31 three-point lines, so 15 is not a sporadic exception to the Burr–Grünbaum–Sloane formula ⌊n(n − 3)/6⌋ + 1 (the known exceptions are n = 7, 11, 16, 19). We also confirm, by the same method, the values t₃(13) = 22 and t₃(14) = 26, which had previously only been announced from an unpublished computation. For arrangements of pseudolines (equivalently, rank-3 oriented matroids) the answer is already different at n = 13: there is an arrangement of 13 pseudolines with 23 triple points, so the pseudoline maximum is t₃ᵖˢ(13) = 23 > 22, and the same holds at n = 14, where t₃ᵖˢ(14) = 27 > 26 = t₃(14). The proof reduces the question to the exact geometric realizability of every partial Steiner triple system on 15 points with 32 blocks whose leave is a 9-edge graph with all degrees even. We enumerate these systems up to isomorphism (7,955,290 isomorphism classes, organised by 13 leave types) and show that none has a realization 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 rest. Every rejection is in fact valid over every field of characteristic 0. The computations are cross-validated by two independent methods (an unfiltered algebraic run and a SAT encoding of rank-3 oriented matroids). The authors are listed in alphabetical order and contributed equally.