On a problem of Neumann

TL;DR

This paper proves that all finite projective planes of even order with an orthogonal polarity contain a Fano subplane, expanding the class of planes known to have such subplanes beyond previously known bounds.

Contribution

It establishes a new sufficient condition for the existence of Fano subplanes in finite projective planes, specifically those of even order with an orthogonal polarity.

Findings

Planes of even order with orthogonal polarity contain Fano subplanes

Number of such planes grows faster than any polynomial in order

Extends known classes of planes with Fano subplanes

Abstract

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 $n$.