@article{KamčevSpiegel, author = {Kamčev, Nina and Spiegel, Christoph}, title = {Another note on intervals in the Hales-Jewett theorem}, series = {Electronic Journal of Combinatorics}, volume = {29}, journal = {Electronic Journal of Combinatorics}, number = {1}, doi = {10.37236/6730}, pages = {P1.62}, abstract = {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 R{\"a}ty for r=2.}, language = {en} }