Improved bounds on the Hadwiger-Debrunner numbers

Let $HD_d(p,q)$ denote the minimal size of a transversal that can always be guaranteed for a family of compact convex sets in $\mathbb{R}^d$ which satisfy the $(p,q)$-property ($p \geq q \geq d+1$). In a celebrated proof of the Hadwiger-Debrunner conjecture, Alon and Kleitman proved that $HD_d(p,q)$ exists for all $p \geq q \geq d+1$. Specifically, they prove that $HD_d(p,d+1)$ is $\tilde{O}(p^{d^2+d})$. We present several improved bounds: (i) For any $q \geq d+1$, $HD_d(p,q) = \tilde{O}(p^{d \left(\frac{q-1}{q-d}\right)})$. (ii) For $q \geq \log p$, $HD_d(p,q) = \tilde{O}(p+(p/q)^d)$. (iii) For every $\epsilon > 0$ there exists a $p_0 = p_0(\epsilon)$ such that for every $p \geq p_0$ and for every $q \geq p^{\frac{d-1}{d}+\epsilon}$ we have: $p-q+1 \leq HD_d(p,q) \leq p-q+2$. The latter is the first near tight estimate of $HD_d(p,q)$ for an extended range of values of $(p,q)$ since the 1957 Hadwiger-Debrunner theorem. We also prove a $(p,2)$-theorem for families in $\mathbb{R}^2$ with union complexity below a specific quadratic bound. Based on this, we introduce a polynomial time constant factor approximation algorithm for MAX-CLIQUE of intersection graphs of convex sets satisfying this property.