Extremal problems for a ne cubes of integers

A collection H of integers is called an aane d-cube if there exist d + 1 positive integers x 0 ; x 1) : We address both density and Ramsey-type questions for aane d-cubes. Regarding density results, upper bounds are found for the size of the largest subset of f1; 2; : : : ; ng not containing an aane d-cube. In 1892 Hilbert published the rst Ramsey-type result for aane 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 f1; 2; : : : ; ng, there is a monochromatic aane d-cube. Improvements for upper and lower bounds on h(d; r) are given for d > 2.