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

For every $k>3$, we give a construction of planar point sets with many collinear $k$-tuples and no collinear $(k+1)$-tuples. We show that there are $n_0=n_0(k)$ and $c=c(k)$ such that if $n\geq n_0$, then there exists a set of $n$ points in the plane that does not contain $k+1$ points on a line, but it contains at least $n^{2-\frac{c}{\sqrt{\log n}}}$ collinear $k$-tuples of points. Thus, we significantly improve the previously best known lower bound for the largest number of collinear $k$-tuples in such a set, and get reasonably close to the trivial upper bound $O(n^2)$.

Many Collinear $$k$$ k -Tuples with no $$k+1$$ k + 1 Collinear Points