On a problem of Neumann

A conjecture widely attributed to Neumann is that all finite non-desarguesian pro-jective 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. Mathematics Subject Classification: 05C99, 51A35, 51A45