A combinatorial proof of the Dense Hindman’s Theorem

The Dense Hindman’s Theorem states that, in any finite coloring of the natural numbers, one may find a single color and a “dense” set B1B1, for each b1∈B1b1∈B1 a “dense” set B2b1 (depending on b1b1), for each b2∈B2b1 a “dense” set B3b1,b2 (depending on b1,b2b1,b2), and so on, such that for any such sequence of bibi, all finite sums belong to the chosen color. (Here density is often taken to be “piecewise syndetic”, but the proof is unchanged for any notion of density satisfying certain properties.) This theorem is an example of a combinatorial statement for which the only known proof requires the use of ultrafilters or a similar infinitary formalism. Here we give a direct combinatorial proof of the theorem.