Counting patterns in colored orthogonal arrays

TL;DR

This paper establishes a combinatorial identity linking color patterns in orthogonal arrays to color class sizes, with applications including bounds on monochromatic triples and dependencies in orthogonal arrays.

Contribution

It introduces a new combinatorial identity relating color patterns and class sizes in orthogonal arrays, with several applications and bounds.

Findings

Every equitable r-coloring of [1,n] has at least 1/2(n/r)^2 + O(n) monochromatic Schur triples.

In OA(d,d-1), monochromatic vector counts depend only on vectors missing that color and class size.

The identity provides a new tool for analyzing colorings in 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.