Some new block designs of dimension three

TL;DR

This paper investigates the rare class of block designs with dimension three, providing explicit constructions, nonexistence results, and applications to triple systems and Latin squares.

Contribution

It introduces new constructions and nonexistence results for dimension three designs with specific block sizes, expanding understanding of their structure and applications.

Findings

Explicit constructions of dimension three designs with block sizes {3,4} and {3,5}

Nonexistence result for certain designs with block sizes {3,5}

Applications to dimension three triple systems and symmetric Latin squares

Abstract

The dimension of a block design is the maximum positive integer $d$ such that any $d$ of its points are contained in a proper subdesign. Pairwise balanced designs PBD$(v,K)$ have dimension at least two as long as not all points are on the same line. On the other hand, designs of dimension three appear to be very scarce. We study designs of dimension three with block sizes in $K=\{3,4\}$ or $\{3,5\}$, obtaining several explicit constructions and one nonexistence result in the latter case. As applications, we obtain a result on dimension three triple systems having arbitrary index as well as symmetric latin squares which are covered in a similar sense by proper subsquares.