We study finite and infinite entangled graphs
in the bond percolation process in three dimensions
with density $p$.
After a discussion of the relevant definitions,
the entanglement critical probabilities are defined.
The size of the maximal entangled graph at the origin
is studied for small $p$, and it is shown that this
graph has radius whose tail decays at least as fast
as $\exp(-\alpha n/\log n)$; indeed, the logarithm
may be replaced by any iterate of logarithm
for an appropriate positive constant $\alpha$. We
explore the question of almost sure uniqueness of
the infinite maximal open entangled graph when $p$
is large, and we establish two relevant theorems.
We make several conjectures concerning the properties
of entangled graphs in percolation. http://www.statslab.cam.ac.uk/$\sim$grg/
1991 Mathematics Subject Classification: primary 60K35;
secondary 05C10, 57M25, 82B41, 82B43, 82D60.