  The tree forcing method given by (Liu 2015) enables the cone avoiding of
strong enumeration of a given tree, within a subset or co-subset of an
arbitrary given set, provided the given tree does not admit computable strong
enumeration. Using this result, we settled and reproduced a series of problems
in reverse mathematics. In this paper, we demonstrate cone avoiding results
within an infinite subset of a given 1-random set. We show that for any given
1-random set $X$, there exists an infinite subset $Y$ of $X$ such that $Y$ does
not compute any real with positive effective Hausdorff dimension, thus
answering negatively a question posed by Kjos-Hanssen that whether there exists
a 1-random set of which any infinite subset computes some 1-random real. The
result is surprising in that the tree forcing technique used on the subset or
co-subset seems to heavily rely on subset co-subset combinatorics, whereas this
result does not.
