An improved bound on the largest induced forests for triangle-free planar graphs

We proved that every planar triangle-free graph of order n has a subset of vertices that induces a forest of size at least (71n + 72)/128. This improves the earlier work of Salavatipour (2006). We also pose some questions regarding planar graphs of higher girth

Induced trees in triangle-free graphs

We prove that every connected triangle-free graph on n vertices contains an induced tree on exp(c √ log n) vertices, where c is a positive constant. The best known upper bound is (2+o(1)) √ n. This partially answers questions of Erdős, Saks, and Sós and of Pultr.