2026/07/28 by Oliver Roche-Newton
Mathematics · #math.CO
For all δ>0 and infinitely many n ∈ \mathbb N, we show that there exists a set L of n lines in \mathbb R2 such that there are no intersecting quadruples, but for every subset L' ⊂ L such that |L'| ≥ n(4)/(5)+δ, there exist three lines from L' with a common point of intersection. This gives an improved bound for a dual form of a theorem of Balogh and Solymosi. As a consequence, we derive an improved lower bound for the Hadwiger-Debrunner number HD2(p,3). We also give, for all 0 ≤ s ≤ 1 and arbitrarily large n ∈ \mathbb N, a construction of a point set S ⊂ [n]3 with cardinality |S|≥ n3-s, such that S contains O(n6-4s) collinear triples. This shows that a supersaturation lemma of Balogh and Solymosi is optimal, up to logarithmic factors.