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

Recently Kostochka, Mubayi and Verstra\"ete initiated the study of the Ramseynumbers of uniform loose cycles versus cliques. In particular they proved that$R(C^r_3,K^r_n) = \tilde{\theta}(n^{3/2})$ for all fixed $r\geq 3$. For thecase of loose cycles of length five they proved that$R(C_5^r,K_n^r)=\Omega((n/\log n)^{5/4})$ and conjectured that $R(C^r_5,K_n^r)= O(n^{5/4})$ for all fixed $r\geq 3$. Our main result is that $R(C_5^3,K_n^3)= O(n^{4/3})$ and more generally for any fixed $l\geq 3$ that $R(C_l^3,K_n^3) =O(n^{1 + 1/\lfloor(l+1)/2 \rfloor})$.  We also explain why for every fixed $l\geq 5$, $r\geq 4$, $R(C^r_l,K^r_n) =O(n^{1+1/\lfloor l/2 \rfloor})$ if $l$ is odd, which improves upon the resultof Collier-Cartaino, Graber and Jiang who proved that for every fixed $r\geq3$, $l\geq 4$, we have $R(C_l^r,K_n^r) = O(n^{1 + 1/(\lfloor l/2 \rfloor-1)})$.

The Ramsey number of loose cycles versus cliques