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We call a $2$-design ${\cal D}$ with parameters $v,k$, and
$\lambda={v-2 \choose k-2}$ a complete design. The number of
distinct blocks of ${\cal D}$, called the support size of ${\cal
D}$, is denoted by $b^*$. For a complete design with $v\geq 7$ and
for $k=3$, Constantine and Hedayat (J. Statist. Plann. Inference
{\bf 7} (1993), 289-294) have shown that max $b^*={v \choose
3}-4(v-3)$, provided a block of ${\cal D}$ attains the maximum
multiplicity, $\lambda$. In this paper, we show that if a block of
a complete design ${\cal D}$ with $k\geq 3$ is repeated maximum
possible times $(i.e.,\lambda= {v-2\choose k-2})$, then $b^*\leq
{v\choose k}-k \left[ {v-1 \choose k-1}-{v-2 \choose
k-2}-{v-k\choose k-1}\right]- {v-2\choose k-2}+1$. Furthermore,
for $v=k^2-k+1$, where $k-1$ is a prime power, and also for
$v\equiv 1$ (mod 12), where $k=4$, we construct designs for which
the equality for $b^*$ holds.
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