Triangle-free induced subgraphs of the unitary polarity graph

Let ⊥ be a unitary polarity of a finite projective plane π of order q ². The unitary polarity graph is the graph with vertex set the points of π where two vertices x and y are adjacent if x∈y ^⊥. We show that a triangle-free induced subgraph of the unitary polarity graph of an arbitrary projective plane has at most (q ⁴+q)∕2 vertices. When π is the Desarguesian projective plane PG(2,q ²) and q is even, we show that the upper bound is asymptotically sharp, by providing an example on q ⁴∕2 vertices. Finally, the case when π is the Figueroa plane is discussed.