# From a $(p, 2)$-Theorem to a Tight $(p, q)$-Theorem Academic Article

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### abstract

• Abstract: 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 …

### publication date

• December 12, 2017