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

We study a Tur\'an-type problem on edge-colored complete graphs. We show thatfor any $r$ and $t$, any sufficiently large $r$-edge-colored complete graph on$n$ vertices with $\Omega(n^{2-1/tr^r})$ edges in each color contains a memberfrom certain finite family $\mathcal{F}_t^r$ of $r$-edge-colored completegraphs. We conjecture that $\Omega(n^{2-1/t})$ edges in each color aresufficient to find a member from ${\mathcal{F}}_t^r$. A result of Gir\~ao andNarayanan confirms this conjecture when $r=2$.  Next, we study a related problem where the corresponding Tur\'an threshold islinear. We call an edge-coloring of a path $P_{rk}$ balanced if each colorappears $k$ times in the coloring. We show that any $3$-edge-coloring of alarge complete graph with $kn+o(n)$ edges in each color contains a balanced$P_{3k}$. This is tight up to a constant factor of $2$. For more colors, theproblem becomes surprisingly more delicate. Already for $r=7$, we show thateven $n^{2-o(1)}$ edges from each color does not guarantee existence of abalanced $P_{7k}$.

Colored unavoidable patterns and balanceable graphs