  The Dense Hindman's Theorem states that, in any finite coloring of the
integers, one may find a single color and a "dense" set $B_1$, for each $b_1\in
B_1$ a "dense" set $B_2^{b_1}$ (depending on $b_1$), for each $b_2\in
B_2^{b_1}$ a "dense" set $B_3^{b_1,b_2}$ (depending on $b_1,b_2$), and so on,
such that for any such sequence of $b_i$, 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.
