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

The Hales–Jewett Theorem states that any $r$–colouring of $[m]^n$ contains a monochromatic combinatorial line if $n$ is large enough. Shelah's proof of the theorem implies that for $m = 3$ there always exists a monochromatic combinatorial line whose set of active coordinates is the union of at most $r$ intervals. For odd $r$, Conlon and Kamčev constructed $r$–colourings for which it cannot be fewer than $r$ intervals. However, we show that for even $r$ and large $n$, any $r$–colouring of $[3]^n$ contains a monochromatic combinatorial line whose set of active coordinates is the union of at most $r-1$ intervals. This is optimal and extends a result of Leader and Räty for $r=2$.

Another Note on Intervals in the Hales–Jewett Theorem