  We provide two ways to show that the R. Thompson group $F$ has maximal
subgroups of infinite index which do not fix any number in the unit interval
under the natural action of $F$ on $(0,1)$, thus solving a problem by D.
Savchuk. The first way employs Jones' subgroup of the R. Thompson group $F$ and
leads to an explicit finitely generated example. The second way employs
directed 2-complexes and 2-dimensional analogs of Stallings' core graphs, and
gives many implicit examples. We also show that $F$ has a decreasing sequence
of finitely generated subgroups $F>H_1>H_2>...$ such that $\cap H_i=\{1\}$ and
for every $i$ there exist only finitely many subgroups of $F$ containing $H_i$.
