On 3-nets in projective planes

Aart Blokhuis

A 3-net in a projective plane is a configuration of $k$ (red) plus $k$ (white) plus $k$ (blue) lines, such that there are $k^2$ intersection points, each on exactly one line of each color. We will describe the known examples, give a characterization in case $k$ is less than characteristic of the field, if the field is finite and just finite if the field is infinite of the case where one of the color classes is a pencil. If time allows we will give the relation of this problem with the Strong Cylinder Conjecture.

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