A family $F$ of sets is said to satisfy the $(p,q)$-property if among any $p$ sets of $F$ some $q$ intersect. The celebrated $(p,q)$-theorem of Alon and Kleitman asserts that any family of compact convex sets in $\mathbb{R}^d$ that satisfies the $(p,q)$-property for some $q \geq d+1$, can be pierced by a fixed number $f_d(p,q)$ of points. The minimum such piercing number is denoted by $HD_d(p,q)$. Already in 1957, Hadwiger and Debrunner showed that whenever $q>\frac{d-1}{d}p+1$ the piercing number is $HD_d(p,q)=p-q+1$; no exact values of $HD_d(p,q)$ were found ever since. While for an arbitrary family of compact convex sets in $\mathbb{R}^d$, $d \geq 2$, a $(p,2)$-property does not imply a bounded piercing number, such bounds were proved for numerous specific families. The best-studied among them is axis-parallel rectangles in the plane. Wegner and (independently) Dol'nikov used a $(p,2)$-theorem for axis-parallel rectangles to show that $HD_{\mathrm{rect}}(p,q)=p-q+1$ holds for all 查看全文>>