In 1974 László Fejes Tóth posed the following problem: place n points in the plane so as to minimise the largest distance from a line meeting the unit-radius disc to the nearest point. Writing r_n for the optimum, he proved r_1 = r_2 = 1 and r_3 = 3/5 exactly, gave constructions showing r_4 ≤ 0.471…, r_5 ≤ 0.406… and r_6 ≤ 1/3, and wrote that r_n is unknown for n > 3. The problem was later named the point goalie problem; its modern literature treats the asymptotic and density regimes, and I am not aware of any published improvement of the finite-n values. This note certifies, in exact rational interval arithmetic, r_4 ≤ 0.468672 and r_5 ≤ 0.394954, improving the 1974 bounds by 0.0023 and 0.0106 respectively. The certifying configurations are given by exact rational coordinates, and the certifier is validated by negative controls against Fejes Tóth's proven value r_3 = 3/5. For n = 6 an unstructured search converged back to Fejes Tóth's own configuration and produced no improvement; that is reported as a negative result, not as evidence of optimality. Finally, Fejes Tóth's skeleton argument applied to the certified opaque barrier of length 4.799849374678… (10.5281/zenodo.21701081) gives lim sup n·r_n ≤ 2.39992468…, improving the asymptotic upper bound (π + √3)/2 = 2.43682… stated in his paper; the best known asymptotic lower bound, due to Richardson and Shepp, is 1.001. No optimality is claimed for anything presented here.
Gonzalez, V. (2026). Certified upper bounds for Fejes Tóth's point-goalie problem at n = 4 and 5. Zenodo. https://doi.org/10.5281/zenodo.21729548