  A conjecture widely attributed to Neumann is that all finite non-desarguesian
projective planes contain a Fano subplane. In this note, we show that any
finite projective plane of even order which admits an orthogonal polarity
contains a Fano subplane. The number of planes of order less than $n$
previously known to contain a Fano subplane was $O(\log n)$, whereas the number
of planes of order less than $n$ that our theorem applies to is not bounded
above by any polynomial in $n$.
