  For a finite measure $\varLambda$ on $[0,1]$, the $\varLambda$-coalescent is
a coalescent process such that, whenever there are $b$ clusters, each $k$-tuple
of clusters merges into one at rate
$\int_0^1x^{k-2}(1-x)^{b-k}\varLambda(\mathrm{d}x)$. It has recently been shown
that if $1<\alpha<2$, the $\varLambda$-coalescent in which $\varLambda$ is the
$\operatorname {Beta}(2-\alpha,\alpha)$ distribution can be used to describe
the genealogy of a continuous-state branching process (CSBP) with an
$\alpha$-stable branching mechanism. Here we use facts about CSBPs to establish
new results about the small-time asymptotics of beta coalescents. We prove an
a.s. limit theorem for the number of blocks at small times, and we establish
results about the sizes of the blocks. We also calculate the Hausdorff and
packing dimensions of a metric space associated with the beta coalescents, and
we find the sum of the lengths of the branches in the coalescent tree, both of
which are determined by the behavior of coalescents at small times. We extend
most of these results to other $\varLambda$-coalescents for which $\varLambda$
has the same asymptotic behavior near zero as the $\operatorname
{Beta}(2-\alpha,\alpha)$ distribution. This work complements recent work of
Bertoin and Le Gall, who also used CSBPs to study small-time properties of
$\varLambda$-coalescents.
