Extremal Problems for Affine Cubes of Integers

A collection H of integers is called an affine d-cube if there exist d+ 1 positive integers x0, x1, . . . , xd so that H = { x0 + ∑ i∈I xi : I ⊆ {1, 2, . . . , d} } . We address both density and Ramsey-type questions for affine d-cubes. Regarding density results, upper bounds are found for the size of the largest subset of {1, 2, . . . , n} not containing an affine d-cube. In 1892 Hilbert published the first Ramsey-type result for affine d-cubes by showing that for any positive integers r and d, there exists a least number n = h(d, r) so that for any r-coloring of {1, 2, . . . , n}, there is a monochromatic affine d-cube. Improvements for upper and lower bounds on h(d, r) are given for d > 2.