1 or at least $(s-2)/3$ empty cells in each row or column. In the case of one empty cell, we correct some results of Wang and Du, and show that there exists a ${\mathrm{GHD}}(n+1,3n)$ if and only if $n \geq 6$, except possibly for $n=6$. In the case of two empty cells, we show that there exists a ${\mathrm{GHD}}(n+2,3n)$ if and only if $n \geq 6$. Noting that the proportion of cells in a given row or column of a ${\mathrm{GHD}}(s,v)$ which are empty falls in the interval $[0,1/3)$, we prove that for any $\pi \in [0,5/18]$, there is a ${\mathrm{GHD}}(s,v)$ whose proportion of empty cells in a row or column is arbitrarily close to $\pi$.
author:
- 'R. Julian R. Abel'
- 'Robert F. Bailey'
- 'Andrea C. Burgess'
- Peter Danziger
- Eric Mendelsohn
title: On generalized Howell designs with block size three
---

[**Keywords:**]{} Generalized Howell designs, triple systems, doubly resolvable designs.

[**MSC2010:**]{} Primary: 05B07, 05B15; Secondary: 05B40, 94B25

Introduction
============

Combinatorial designs on square arrays have been the subject of much attention, with mutually orthogonal Latin squares being the most natural example. Block designs with two orthogonal resolutions can also be thought of in this way, 