Counting patterns in colored orthogonal arrays

Abstract

Let S be an orthogonal array OA(d,k) and let c be an r-coloring of its ground set X. We give a combinatorial identity which relates the number of vectors in S with given color patterns under c with the cardinalities of the color classes. Several applications of the identity are considered. Among them it is shown that every coloring of an orthogonal array OA(d,d−1) contains a positive proportion of almost rainbow vectors and also of almost monochromatic vectors of every color.