7 research outputs found
Pairwise balanced designs covered by bounded flats
We prove that for any and , there exist, for all sufficiently large
admissible , a pairwise balanced design PBD of dimension for
which all -point-generated flats are bounded by a constant independent of
. We also tighten a prior upper bound for , 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
Mutually orthogonal latin squares with large holes
Two latin squares are orthogonal if, when they are superimposed, every
ordered pair of symbols appears exactly once. This definition extends naturally
to `incomplete' latin squares each having a hole on the same rows, columns, and
symbols. If an incomplete latin square of order has a hole of order ,
then it is an easy observation that . More generally, if a set of
incomplete mutually orthogonal latin squares of order have a common hole of
order , then . In this article, we prove such sets of
incomplete squares exist for all satisfying
Asymptotic existence theorems for frames and group divisible designs
AbstractIn this paper, we establish an asymptotic existence theorem for group divisible designs of type mn with block sizes in any given set K of integers greater than 1. As consequences, we will prove an asymptotic existence theorem for frames and derive a partial asymptotic existence theorem for resolvable group divisible designs