Coloring \(t\)-dimensional \(m\)-boxes (Q1841894)

Results related to Ramsey theory and to discrepancy theory are discussed for colorings of \(t\)-dimensional grids. In particular, let \(R_t(m,r)\) denote the smallest integer \(R\) such that every \(r\)-coloring of the \(t\)-fold Cartesian product of \([R]=\{1,\ldots,R\}\) contains a monochromatic \(t\)-dimensional \(m\)-box. Lower and upper bounds for \(R_t(m,r)\) are derived in the first part of the paper, where the lower bound is obtained using the probabilistic method. The second part of the paper considers the discrepancy of the two-dimensional \(m\)-boxes for which also lower and upper bounds are given.