# On Piercing Numbers of Families Satisfying the $(p, q) _r$ Property Academic Article

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### abstract

• The Hadwiger-Debrunner number $HD_d (p, q)$ is the minimal size of a piercing set that can always be guaranteed for a family of compact convex sets in $\mathbb {R}^ d$ that satisfies the $(p, q)$ property. Hadwiger and Debrunner showed that $HD_d (p, q)\geq p- q+ 1$ for all $q$, and equality is attained for $q>\frac {d-1}{d} p+ 1$. Almost tight upper bounds for $HD_d (p, q)$ for asufficiently large'$q$ were obtained recently using an enhancement of the celebrated Alon-Kleitman theorem, but no sharp upper bounds for a general $q$ are known. In [L. Montejano and P. Sober\'{o} n, Piercing numbers for balanced and unbalanced families, Disc. Comput. Geom., 45 (2)(2011), pp. 358-364], Montejano and Sober\'{o} n defined a refinement of the $(p, q)$ property: $F$ satisfies the $(p, q) _r$ property if among any $p$ elements of $F$, at least $r$ of the $q$-tuples …

### publication date

• March 18, 2017