Triangle-free planar graphs with small independence number

Since planar triangle-free graphs are 3-colourable, such a graph with n vertices has an independent set of size at least n∕3. We prove that unless the graph contains a certain obstruction, its independence number is at least n∕(3−ε) for some fixed ε>0. We also provide a reduction rule for this obstruction, which enables us to transform any plane triangle-free graph G into a plane triangle-free graph G′ such that α(G′)−|G′|∕3=α(G)−|G|∕3 and|G′|≤(α(G)−|G|∕3)∕ε. We derive a number of algorithmic consequences as well as a structural description of n-vertex plane triangle-free graphs whose independence number is close to n∕3.