  We consider various lattice models of polymers: lattice trees, lattice
animals, and self-avoiding walks. The polymer interacts with a surface
(hyperplane), receiving a unit energy reward for each site in the surface.
There is an adsorption transition of the polymer at a critical value of
$\beta$, the inverse temperature. We present a new proof of the result of
Hammersley, Torrie, and Whittington (1982) that the transition occurs at a
strictly positive value of $\beta$ when the surface is impenetrable, i.e. when
the polymer is restricted to a half-space. In contrast, for a penetrable
surface, it is an open problem to prove that the transition occurs at $\beta=0$
(i.e., infinite temperature). We reduce this problem to showing that the
fraction of N-site polymers whose span is less than $N/\log^2 N$ is not too
small.
