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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 $\\pi$ is the Desarguesian projective plane $\\mathrm{PG}(2,q^2)$ and $q$ is even, we show that the upper bound is asymptotically sharp, by providing an example on $q^4/2$ vertices. 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