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### abstract

• Abstract: 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 $\tilde {O}(p^{d^ 2+ d})$.

### publication date

• December 13, 2015