  The evolution of the usual Erd\H{o}s-R\'{e}nyi random graph model on n
vertices can be described as follows: At time 0 start with the empty graph,
with n vertices and no edges. Now at each time k, choose 2 vertices uniformly
at random and attach an edge between these two vertices. Let \bfG_n(k) be the
graph obtained at step k. Refined analysis in random graph theory now shows
that for fixed t\in \Rbold, when k(n) = n/2+ n^{2/3} t/2, the sizes of the
components in \bfG_n(k(n)) scale like n^{2/3} and rescaled component sizes
converge to the standard multiplicative coalescent at time $t$. The last decade
has seen variants of this process introduced, under the name Achlioptas
processes, to understand the effect of simple changes in the edge formation
scheme on the emergence of the giant component. Stimulated by a question of
Achlioptas, one of the simplest and most popular of such models is the Bohman
Frieze (BF) model wherein at each stage $k$, 2 edges e_1(k)=(v_1,v_2) and
e_2(k) = (v_3, v_4) are chosen uniformly at random. If at this time v_1, v_2
are both isolated then this edge is added, otherwise e_2 is added. Then
\cite{bohman2001avoiding} (and further analysis in \cite{spencer2007birth})
show that once again there is a critical parameter, which is larger than 1,
above and below which the asymptotic behavior is as in the Erd\H{o}s-R\'{e}nyi
setting. While an intense study for this and related models seems to suggest
that at criticality, this model should be in the same universality class as the
original Erd\H{o}s-R\'{e}nyi process, a precise mathematical treatment of the
dynamics in the critical window has to date escaped analysis. In this work we
study the component structure of the BF model in the critical window and show
that at criticality the sizes of components properly rescaled and re-centered
converge to the standard multiplicative coalescent.
