A new lower bound on Hadwiger-Debrunner numbers in the plane

A family of sets $${\cal F}$$ is said to satisfy the (p, q) property if among any p in some q have a non-empty intersection. Hadwiger and Debrunner (1957) conjectured that for ≥ d + 1 there exists minimum integer c = HDd(p, q), such finite convex ℝd satisfies can be pierced by at most points. In celebrated result from 1992, Alon Kleitman proved conjecture. However, obtaining sharp bounds on known as ‘the Hadwiger-Debrunner numbers’, still major open problem discrete computational geometry. The best upper bound numbers plane $$O({p^{(1.5 \delta)(1 {1 \over {q - 2}})}})$$ (for δ > 0 q0(δ)), obtained combining results Keller, Smorodinsky Tardos (2017) Rubin (2018). lower $$H\,{D_2}(p,q) \Omega \left({{p q}\log q}} \right)} \right)$$ , Bukh, Matoušek Nivasch more than 10 years ago. this paper we improve significantly showing HD2(p, p1+Ω(1/q). Furthermore, lines tight all families bounded VC-dimension. Unlike previous numbers, which mainly used weak epsilon-net theorem, our stems surprising connection an old Erdős points general position plane. We use novel construction Erdős’ problem, recently Balogh Solymosi using hypergraph container method, get 3). then generalize 3.