Counting patterns in colored orthogonal arrays
Amanda Montejano, Oriol Serra

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 be an orthogonal array and let be an --coloring of its ground set . We give a combinatorial identity which relates the number of vectors in with given color patterns under with the cardinalities of the color classes. Several applications of the identity are considered. Among them, we show that every equitable --coloring of the integer interval has at least monochromatic Schur triples. We also show that in an orthogonal array , 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.
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Taxonomy
Topicsgraph theory and CDMA systems · Limits and Structures in Graph Theory · Point processes and geometric inequalities
