For a pla­nar graph $G$, a free set is a sub­set $F\subset V(G)$ such that, for any set $X$ of $|F|$ points in $\mathbb{R}^2$, there is a non-cross­ing straight-line draw­ing of $G$ in which the points in $F$ are drawn on the points of $X$. The ex­is­tence of large free sets in $n$-ver­tex pla­nar graphs has been used to give the tight­est re­sults for a num­ber of prob­lems in graph draw­ing and geo­met­ric graph the­ory in­clud­ing un­tan­gling, col­umn pla­narity, uni­ver­sal point sub­sets, and par­tial si­mul­ta­ne­ous geo­met­ric draw­ings. For the last ten years, progress on these prob­lems has been lim­ited by the fact that the best known lower bound on the size of free sets in $n$-ver­tex pla­nar graphs is $\Omega(n^{0.5})$.

In this talk, I will show that if $G$ is a tri­an­gu­la­tion of max­i­mum de­gree $\Delta$ and the dual of $G$ con­tains a sim­ple cycle of length $\ell$, then $G$ has a free set $F$ of size $\Omega(\ell/\Delta^4)$. This, along with the cur­rent best lower-bounds on the cir­cum­fer­ence of 3-reg­u­lar 3-con­nected pla­nar graphs shows that every $n$-ver­tex pla­nar graph of max­i­mum de­gree $\Delta$ has a free set of size $\Omega(n^{0.8}/\Delta^4)$. This gives im­proved re­sults for all the ap­pli­ca­tions men­tioned above in the class of pla­nar graphs of bounded de­gree.