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