Hocquard, Kim, and Pierron constructed, for every even integer D≥2, a 2-degenerate graph GD with maximum degree D such that ω(GD2)=25D
Nearby in the stack
. We prove for (a) all 2-degenerate graphs
G
and (b) all graphs
G
with
mad(G)<4
, upper bounds on the clique number
ω(G2)
of
G2
that match the lower bound given by this construction, up to small additive constants. We show that if
G
is 2-degenerate with maximum degree
D
, then
ω(G2)≤25D+72
(with
ω(G2)≤25D+60
when
D
is sufficiently large). And if
G
has
mad(G)<4
and maximum degree
D
, then
ω(G2)≤25D+532
. Thus, the construction of Hocquard et al. is essentially best possible. Our proofs introduce a "token passing" technique to derive crucial information about non-adjacencies in
G
of vertices that are adjacent in
G2
. This is a powerful technique for working with such graphs that has not previously appeared in the literature.