Before revision:  The best currently known asymptotic lower bound on the Hadwiger-Debrunner numbers in the plane is c_2(p,q) = \Omega( p{q}\log(p{q}))  . In this paper we present the significantly stronger lower bound c_2(p,q) \geq p^{1+\Omega(1/q)}.
After revision:  The best currently known asymptotic lower bound on the Hadwiger-Debrunner numbers in the plane is c_2(p,q) = \Omega( p{q}\log(p{q})) while the best known upper bound is O(p^{(1.5+\delta)(1+\frac{1 . In this paper we present the significantly stronger lower bound c_2(p,q) \geq p^{1+\Omega(1/q)}.