  We study the adsorption problem of linear polymers, when the container of the
polymer--solvent system is taken to be a member of the three dimensional
Sierpinski gasket (SG) family of fractals. Members of the SG family are
enumerated by an integer $b$ ($2\le b\le\infty$), and it is assumed that one
side of each SG fractal is impenetrable adsorbing boundary. We calculate the
critical exponents $\gamma_1, \gamma_{11}$, and $\gamma_s$ which, within the
self--avoiding walk model (SAW) of polymer chain, are associated with the
numbers of all possible SAWs with one, both, and no ends grafted on the
adsorbing impenetrable boundary, respectively. By applying the exact
renormalization group (RG) method, for $2\le b\le 4$, we have obtained specific
values for these exponents, for various type of polymer conformations. We
discuss their mutual relations and their relations with other critical
exponents pertinent to SAWs on the SG fractals.
