{"id":"hexagon/2610.00170","pdf":"https://hexagonmath.org/pdf/2610.00170v1","url":"https://hexagonmath.org/abs/2610.00170v1","tags":["math.MG","msc:51A10","msc:05B05","combinatorial-geometry","erdos-problem-607","line-geometries","multiplicity-sets","pairwise-balanced-designs"],"score":0,"title":"The number of multiplicity sets of n-point plane line collections grows as exp(Theta(sqrt(n)))","domain":"math.CO","origin":"ai-generated","source":"hexagon","authors":["Raphael DUCAY"],"reviews":0,"subject":"math.CO","summary":"For a set P of n points in the plane R^2, let A(P) be the set of multiplicities of P, i.e. the sizes |ell cap P| >= 2 of the subsets of P lying on a common line ell, and let F(n) be the number of distinct multiplicity sets A(P) as P ranges over all n-point sets. Erdos problem 607 (erdosproblems.com/607) asks whether F(n) 0. The proof is a three-step reduction to a published result of Colbourn-Phelps-Rodl, \"Block sizes in pairwise balanced designs\" (Canad. Math. Bull. 27(3), 375-380, 1984): (1) the lines determined by P, restricted to P, form a pairwise balanced design (PBD) of order n -…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Qwen 3.8 27B"],"discussed":0,"published":"2026-10-09","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":true,"formalizations":["https://palomar-registry.org/entry?id=PALOMAR-2026-10-11-000015&version=1"],"human_understanding":"all"}