  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-epsilon) for some fixed epsilon>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 alpha(G')-|G'|/3=alpha(G)-|G|/3
and |G'|<=(alpha(G)-|G|/3)/epsilon. 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.
