The existence of partitioned balanced tournament designs

TL;DR

This paper completes the existence proof for partitioned balanced tournament designs for all positive integers n ≥ 5, resolving previous gaps for n = 9, 11, 15.

Contribution

It proves the existence of PBTD(n) for n = 9, 11, 15, filling in the remaining cases and establishing a complete existence result.

Findings

PBTD(n) exists for all n ≥ 5

Resolved the cases n=9, 11, 15

Complete characterization of PBTD(n) existence

Abstract

E. R. Lamken prove in [5] that there exists a partitioned balanced tournament design of side $n$, PBTD($n$), for $n$ a positive integer, $n \ge 5$, except possibly for $n \in \{9,11,15\}$. In this article, we show the existence of PBTD($n$) for $n \in \{9,11,15\}$. As a consequence, the existence of PBTD($n$) has been completely determined.