On colorings of bivariate random sequences

The ergodic sequences consisting of vectors (ξn, ηn), n ⩾ 1, over a finite alphabet A×B are colored with ⌊e^nα⌋ colors for Aⁿ and⌊e^nβ⌋ colors for Bⁿ. Generic behavior of the colorings in terms of probabilities of monochromatic rectangles intersected with typical sets is examined. When n increases a big majority of pairs of colorings produces rectangles whose probabilities are bounded uniformly from above. Limiting rates of bounds are worked out in all regimes of the rates a and β of colorings. As a consequence, generic behavior of the colorings in terms of Shannon entropies of the partitions into rectangles is described.