Subsets of Fq[x] free of 3-term geometric progressions

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 Fq[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 Fq[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 positive characteristic introduces some extra combinatorial structure that increases the tractability of common questions in this area.