English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

A range family $\mathcal{R}$ is a family of subsets of $\mathbb{R}^d$, likeall halfplanes, or all unit disks. Given a range family $\mathcal{R}$, weconsider the $m$-uniform range capturing hypergraphs$\mathcal{H}(V,\mathcal{R},m)$ whose vertex-sets $V$ are finite sets of pointsin $\mathbb{R}^d$ with any $m$ vertices forming a hyperedge $e$ whenever $e = V\cap R$ for some $R \in \mathcal{R}$. Given additionally an integer $k \geq 2$,we seek to find the minimum $m = m_{\mathcal{R}}(k)$ such that every$\mathcal{H}(V,\mathcal{R},m)$ admits a polychromatic $k$-coloring of itsvertices, that is, where every hyperedge contains at least one point of eachcolor. Clearly, $m_{\mathcal{R}}(k) \geq k$ and the gold standard is an upperbound $m_{\mathcal{R}}(k) = O(k)$ that is linear in $k$.  A $t$-shallow hitting set in $\mathcal{H}(V,\mathcal{R},m)$ is a subset $S\subseteq V$ such that $1 \leq |e \cap S| \leq t$ for each hyperedge $e$; i.e.,every hyperedge is hit at least once but at most $t$ times by $S$. We show forseveral range families $\mathcal{R}$ the existence of $t$-shallow hitting setsin every $\mathcal{H}(V,\mathcal{R},m)$ with $t$ being a constant onlydepending on $\mathcal{R}$. This in particular proves that $m_{\mathcal{R}}(k)\leq tk = O(k)$ in such cases, improving previous polynomial bounds in $k$.Particularly, we prove this for the range families of all axis-aligned stripsin $\mathbb{R}^d$, all bottomless and topless rectangles in $\mathbb{R}^2$, andfor all unit-height axis-aligned rectangles in $\mathbb{R}^2$.

Polychromatic Colorings of Geometric Hypergraphs via Shallow Hitting Sets