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 Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Several classical constructions illustrate the fact that the chromatic numberof a graph can be arbitrarily large compared to its clique number. However,until very recently, no such construction was known for intersection graphs ofgeometric objects in the plane. We provide a general construction that for anyarc-connected compact set $X$ in $\mathbb{R}^2$ that is not an axis-alignedrectangle and for any positive integer $k$ produces a family $\mathcal{F}$ ofsets, each obtained by an independent horizontal and vertical scaling andtranslation of $X$, such that no three sets in $\mathcal{F}$ pairwise intersectand $\chi(\mathcal{F})>k$. This provides a negative answer to a question ofGyarfas and Lehel for L-shapes. With extra conditions, we also show how toconstruct a triangle-free family of homothetic (uniformly scaled) copies of aset with arbitrarily large chromatic number. This applies to many commonshapes, like circles, square boundaries, and equilateral L-shapes.Additionally, we reveal a surprising connection between coloring geometricobjects in the plane and on-line coloring of intervals on the line.

Triangle-free geometric intersection graphs with large chromatic number