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 $r=2$.
@article{10_37236_9400,
author = {Nina Kam\v{c}ev and Christoph Spiegel},
title = {Another note on intervals in the {Hales-Jewett} theorem},
journal = {The electronic journal of combinatorics},
year = {2022},
volume = {29},
number = {1},
doi = {10.37236/9400},
zbl = {1486.05306},
url = {http://geodesic.mathdoc.fr/articles/10.37236/9400/}
}
TY - JOUR
AU - Nina Kamčev
AU - Christoph Spiegel
TI - Another note on intervals in the Hales-Jewett theorem
JO - The electronic journal of combinatorics
PY - 2022
VL - 29
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/9400/
DO - 10.37236/9400
ID - 10_37236_9400
ER -
%0 Journal Article
%A Nina Kamčev
%A Christoph Spiegel
%T Another note on intervals in the Hales-Jewett theorem
%J The electronic journal of combinatorics
%D 2022
%V 29
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/9400/
%R 10.37236/9400
%F 10_37236_9400
Nina Kamčev; Christoph Spiegel. Another note on intervals in the Hales-Jewett theorem. The electronic journal of combinatorics, Tome 29 (2022) no. 1. doi: 10.37236/9400
