Abstract: | Given integers , a family of sets satisfies the property if among any members of it some intersect. We prove that for any fixed integer constants , a family of -intervals satisfying the property can be pierced by points, with constants depending only on and . This extends results of Tardos, Kaiser and Alon for the case , and of Kaiser and Rabinovich for the case . We further show that similar bounds hold in families of subgraphs of a tree or a graph of bounded tree-width, each consisting of at most connected components, extending results of Alon for the case . Finally, we prove an upper bound of on the fractional piercing number in families of -intervals satisfying the property, and show that this bound is asymptotically sharp. |