  Several recent papers have considered the Ramsey-theoretic problem of how
large a subset of integers can be without containing any 3-term geometric
progressions. This problem has also recently been generalized to number fields,
determining bounds on the greatest possible density of ideals avoiding
geometric progressions. We study the analogous problem over $\mathbb{F}_q[x]$,
first constructing a set greedily which avoids these progressions and
calculating its density, and then considering bounds on the upper density of
subsets of $\mathbb{F}_q[x]$ which avoid 3-term geometric progressions. This
new setting gives us a parameter $q$ to vary and study how our bounds converge
to 1 as it changes, and finite characteristic introduces some extra
combinatorial structure that increases the tractibility of common questions in
this area.
