A note on intervals in the Hales-Jewett theorem (Q1658753)

Summary: The Hales-Jewett theorem for alphabet of size 3 states that whenever the Hales-Jewett cube \([3]^{n}\) is \(r\)-coloured there is a monochromatic line (for \(n\) large). \textit{D. Conlon} and \textit{N. Kamcev} [``Intervals in the Hales-Jewett theorem'', Preprint, \url{arXiv:1801.08919}] conjectured that, for any \(n\), there is a 2-colouring of \([3]^{n}\) for which there is no monochromatic line whose active coordinate set is an interval. In this note we disprove this conjecture.