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, we show that every equitable $r$--coloring of the integer interval $[1,n]$ has at least $1/2(n/r{)}^2+O(n)$ 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.

12 pages

References in corpus (1)

• On the asymptotic minimum number of monochromatic 3-term arithmetic progressions

Cited by in corpus (1)

• Rainbow-free 3-colorings of Abelian Groups