Gallai theorem for an integer lattice
= Gallai theorem for an integer lattice
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= Gallai's theorem
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For every finite $S\subseteq\mathbb N^d$, every <finite coloring> of $\mathbb N^d$ contains a monochromatic homothetic copy $a+rS$. The theorem follows from the <Hales-Jewett theorem> by coloring a word according to the sum of the points indexed by its letters.