  Let $N$ be the number of triangles in an Erd\H{o}s-R\'enyi graph
$\mathcal{G}(n,p)$ on $n$ vertices with edge density $p=d/n,$ where $d>0$ is a
fixed constant. It is well known that $N$ weakly converges to the Poisson
distribution with mean ${d^3}/{6}$ as $n\rightarrow \infty$. We address the
upper tail problem for $N,$ namely, we investigate how fast $k$ must grow, so
that the probability of $\{N\ge k\}$ is not well approximated anymore by the
tail of the corresponding Poisson variable. Proving that the tail exhibits a
sharp phase transition, we essentially show that the upper tail is governed by
Poisson behavior only when $k^{1/3} \log k< (\frac{3}{\sqrt{2}})^{2/3} \log n$
(sub-critical regime) as well as pin down the tail behavior when $k^{1/3} \log
k> (\frac{3}{\sqrt{2}})^{2/3} \log n$ (super-critical regime). We further prove
a structure theorem, showing that the sub-critical upper tail behavior is
dictated by the appearance of almost $k$ vertex-disjoint triangles whereas in
the supercritical regime, the excess triangles arise from a clique like
structure of size approximately $(6k)^{1/3}$. This settles the long-standing
upper-tail problem in this case, answering a question of Aldous, complementing
a long sequence of works, spanning multiple decades, culminating in (Harel,
Moussat, Samotij,'19) which analyzed the problem only in the regime $p\gg
\frac{1}{n}.$ The proofs rely on several novel graph theoretical results which
could have other applications.
