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The usual approach is to induct on the length of the arithmetic progression, which is difficult to simplify directly to the case of two colors.

Does anyone know a different approach?

Thank you.

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1 Answer 1

up vote 2 down vote accepted

There is Shehlah's approach for the Hales Jewett theorem (of which Van der Waerden is a special case). See,'s_proof

Not sure whether it is simpler, though.

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