  Compacta X and Y are said to admit a stable intersection in R^n if there are
maps f : X -> R^n and g : Y -> R^n such that for every sufficiently close
continuous approximations f' : X -> R^n and g' : Y -> R^n of f and g we have
f'(X)\cap g'(Y)\neq\emptyset. The well-known conjecture asserting that X and Y
do not admit a stable intersection in R^n if and only if dim X \times Y \leq
n-1 was confirmed in many cases. In this paper we prove this conjecture in all
the remaining cases except the case dim X =dim Y =3, dim X \times Y=4 and n=5
which still remains open.
