Extremal Problems for Affine Cubes of Integers

Abstract

A collection H of integers is called an affine d-cube if there exist d+1 positive integers x0,x1,…, xd so thatformula hereWe 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-colouring 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.