Another Note on Intervals in the Hales-Jewett Theorem

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 Kamcev 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 Ray for r = 2.