= Sum map from words to homothetic copies
{title2=$\pi(w)=b+\sum_jv_{w_j}$}
For a finite pattern $\{v_1,\ldots,v_k\}$ in a <real vector space>, pull a <finite colouring> back to words through the displayed sum map. A <combinatorial line> with active coordinates J maps to $a+|J|\{v_1,\ldots,v_k\}$. The <Hales-Jewett theorem> therefore forces a <monochromatic> <homothetic copy>. <Integer> patterns remain in the <integer lattice>; choosing b sufficiently large gives the positive-lattice version.
Back to article page
