Counting patterns in colored orthogonal arrays

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, we show that every equitable $r$–coloring of the integer interval $[1,n]$ has at least $1\mathrm{/}2(n\mathrm{/}r{)}^2+O(n)$ monochromatic Schur triples. We also show that in an orthogonal array $OA(d,d-1)$, the number of monochromatic vectors of each color depends only on the number of vectors which miss that color and the cardinality of the color class.