An $ε$-net theorem for a hypergraph upper bounds the minimum size of a vertex set that pierces all $ε$-heavy hyperedges. A $(p,2)$-theorem bounds from above the minimum size of a vertex set that pierces all hyperedges, in terms of the maximum size of a set of pairwise disjoint hyperedges. Numerous works studied $ε$-net theorems and $(p,2)$-theorems that guarantee the existence of small-sized piercing sets. We focus on the question: In which settings the asymptotically smallest possible piercing sets -- i.e., $ε$-nets of size $O(\frac{1}ε)$ and piercing sets of size $O(p)$ in $(p,2)$-theorems, are guaranteed? We obtain several sufficient criteria for the existence of such linear $ε$-net theorems and $(p,2)$-theorems that unveil interesting connections to graph theory and improve and generalize several previous results. Most notably, we exhibit an unexpected relation of $ε$-nets to the classical Zarankiewicz's problem in graph theory. We show that a linear bound in the Zarankiewicz-type problem that asks for the maximum size of a bipartite graph with no copy of $K_{2,t}$, implies a linear $ε$-net theorem for the corresponding neighborhood hypergraph. We also show that hypergraphs with a hereditarily linear-sized Delaunay graph admit an almost linear $(p,2)$-theorem, and deduce that incidence hypergraphs of non-piercing regions in the plane admit a linear $(p,2)$-theorem, significantly improving previous results on such hypergraphs. Our work presents a landscape of sufficient conditions for the existence of linear $ε$-net theorems and $(p,2)$-theorems, with complex interrelations between them. Many of the interrelations are still unknown and call for future research.

翻译：超图的ε网定理给出了刺穿所有ε-重超边所需顶点集最小规模的上界。(p,2)定理则依据两两不交超边集的最大规模，给出了刺穿所有超边所需顶点集最小规模的上界。大量研究致力于探索能保证存在小规模刺穿集的ε网定理与(p,2)定理。我们关注以下问题：在何种设定下能保证存在渐近意义下最小可能的刺穿集——即规模为O(1/ε)的ε网以及(p,2)定理中规模为O(p)的刺穿集？我们获得了线性ε网定理与(p,2)定理存在的若干充分判据，这些判据揭示了与图论的有趣联系，并改进和推广了多项先前结果。尤为值得注意的是，我们发现了ε网与图论中经典Zarankiewicz问题之间出人意料的关联。我们证明，在Zarankiewicz型问题（探究不含K_{2,t}子图的二部图的最大规模）中成立的线性界，意味着对应邻域超图存在线性ε网定理。我们还证明了具有遗传线性规模Delaunay图的超图满足几乎线性的(p,2)定理，并由此推导出平面上非刺穿区域的关联超图满足线性(p,2)定理，这显著改进了此类超图的已有结果。我们的工作呈现了线性ε网定理与(p,2)定理存在性充分条件的整体图景，其中包含复杂的相互关联。许多关联仍有待探索，亟待未来研究。