A combinatorial proof of the Dense Hindman's Theorem

作者：Towsner Henry^*

来源：Discrete Mathematics, 2011, 311(14): 1380-1384.

DOI：10.1016/j.disc.2011.03.006

• 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 B(1), for each b(1) is an element of B(1) a "dense" set B(2)(b1) (depending on b(1)), for each b(2) is an element of B(2)(b1) a "dense" set B(3)(b1.b2) (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.

• 出版日期2011-7-28