Uniformly Resolvable Designs

A \(k\)-URD\((v, g, r)\) is a resolvable design on \(v\) points with block sizes \(g\) and \(k\). Each parallel class contains only one block size, and there are \(r\) parallel classes with blocks of size \(g\), this implies there are \(\frac{v-1-r(g-1)}{k-1}\) parallel classes of size \(k\).

We show that for sufficiently large \(v\), the necessary conditions are sufficient for the following range of values of \(r\). Let \(\epsilon_{k,g} = 1\)if \(g \equiv 0 \mod{k}\) and \(k\) otherwise, and let \(u = \frac{v}{g\epsilon_{k,g}}\).

If \(k = 2\) for all \(g\), or \(k = 3\) with \(g\) odd, then there exists a \(k\)-URD\((v, g, r)\) for the following values of \(r\):

1. If \(u\) is an odd prime power, then for all \(1 \leq r \leq \frac{1}{k-1}(u-1)\), except possibly for the case where \(k = 3\) and \(u\) is congruent to \(5 \mod 6\).
2. If \(u\) is not prime, then for all \(1 \leq r \leq \frac{u}{p}\), where \(p\) is the smallest proper factor of \(u\).