The $\epsilon$-$t$-Net Problem

TLDR: It is proved that any sufficiently large hypergraph with VC-dimension $d admits an $\epsilon$-$t$-net of size $O(\frac{ (1+\log t)d}{\ep silon} \log \frac{1}{\ epsilon})$.

Abstract: We study a natural generalization of the classical $\epsilon$-net problem (Haussler--Welzl 1987), which we call the "$\epsilon$-$t$-net problem": Given a hypergraph on $n$ vertices and parameters $t$ and $\epsilon\geq \frac t n$, find a minimum-sized family $S$ of $t$-element subsets of vertices such that each hyperedge of size at least $\epsilon n$ contains a set in $S$. When $t=1$, this corresponds to the $\epsilon$-net problem. We prove that any sufficiently large hypergraph with VC-dimension $d$ admits an $\epsilon$-$t$-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 $O(\frac{1}{\epsilon})$-sized $\epsilon$-$t$-nets. We also present an explicit construction of $\epsilon$-$t$-nets (including $\epsilon$-nets) for hypergraphs with bounded VC-dimension. In comparison to previous constructions for the special case of $\epsilon$-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.