  The problem of reconstructing a sequence from the set of its length-$k$
substrings has received considerable attention due to its various applications
in genomics. We study an uncoded version of this problem where multiple random
sources are to be simultaneously reconstructed from the union of their $k$-mer
sets. We consider an asymptotic regime where $m = n^\alpha$ i.i.d. source
sequences of length $n$ are to be reconstructed from the set of their
substrings of length $k=\beta \log n$, and seek to characterize the
$(\alpha,\beta)$ pairs for which reconstruction is information-theoretically
feasible. We show that, as $n \to \infty$, the source sequences can be
reconstructed if $\beta > \max(2\alpha+1,\alpha+2)$ and cannot be reconstructed
if $\beta < \max( 2\alpha+1, \alpha+ \tfrac32)$, characterizing the feasibility
region almost completely. Interestingly, our result shows that there are
feasible $(\alpha,\beta)$ pairs where repeats across the source strings abound,
and non-trivial reconstruction algorithms are needed to achieve the fundamental
limit.
