The (p,q) property in families of d-intervals and d-trees

Given integers $p\ge q>1$, a family of sets satisfies the $(p,q)$ property if among any $p$ members of it some $q$ intersect. We prove that for any fixed integer constants $p\ge q>1$, a family of $d$-intervals satisfying the $(p,q)$ property can be pierced by $O\left(d^{\frac{q}{q-1}}\right)$ points, with constants depending only on $p$ and $q$. This extends results of Tardos, Kaiser and Alon for the case $q=2$, and of Kaiser and Rabinovich for the case $p=q=\lceil lo{g}_2(d+2)\rceil$. 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 $d$ connected components, extending results of Alon for the case $q=2$. Finally, we prove an upper bound of $O\left(d^{\frac{1}{p-1}}\right)$ on the fractional piercing number in families of $d$-intervals satisfying the $(p,p)$ property, and show that this bound is asymptotically sharp.