Pairwise balanced designs covered by bounded flats

TL;DR

This paper proves the existence of large pairwise balanced designs with bounded flats for any parameters, and improves bounds for specific cases, enabling constructions like Latin squares covered by small subsquares.

Contribution

It establishes the existence of pairwise balanced designs with bounded flats for all large admissible sizes and refines bounds for certain small block sizes.

Findings

Existence of PBDs with bounded flats for all large v and any K,d

Tightened upper bounds for K={3,4,5}

Constructed Latin squares covered by small subsquares

Abstract

We prove that for any $K$ and $d$, there exist, for all sufficiently large admissible $v$, a pairwise balanced design PBD$(v,K)$ of dimension $d$ for which all $d$-point-generated flats are bounded by a constant independent of $v$. We also tighten a prior upper bound for $K = \{3,4,5\}$, in which case there are no divisibility restrictions on the number of points. One consequence of this latter result is the construction of latin squares `covered' by small subsquares.