The $ϵ$-$t$-Net Problem

We study a natural generalization of the classical $ϵ$-net problem (Haussler–Welzl 1987), which we call the “$ϵ$-$t$-net problem”: Given a hypergraph on $n$ vertices and parameters $t$ and $ϵ\ge \frac{t}{n}$, find a minimum-sized family $S$ of $t$-element subsets of vertices such that each hyperedge of size at least $ϵn$ contains a set in $S$. When $t=1$, this corresponds to the $ϵ$-net problem. We prove that any sufficiently large hypergraph with VC-dimension $d$ admits an $ϵ$-$t$-net of size $O(\frac{ (1+\log t)d}{\epsilon} \log \frac{1}{\epsilon})$. For some families of geometrically-defined hypergraphs (such as the dual hypergraph of regions with linear union complexity), we prove the existence of $O(\frac{1}{ϵ})$-sized $ϵ$-$t$-nets. We also present an explicit construction of $ϵ$-$t$-nets (including $ϵ$-nets) for hypergraphs with bounded VC-dimension. In comparison to previous constructions for the special case of $ϵ$-nets (i.e., for $t=1$), it does not rely on advanced derandomization techniques. To this end we introduce a variant of the notion of VC-dimension which is of independent interest.