I like the three color addition! Is this a common property of projective planes (maybe it's trivial?...I'm not used to thinking about the points as repeated like this! 🙂 ) or one of the named adjectives (Desarguesian, etc.)?
@bmreiniger it's three because on a Fano plane each point lies on three lines. In the projective plane, there'd be as many colours as pictures on a single card (is it 7?)
Sorry, I wasn't clear, by "this" I meant the ability to then "write down" all the lines using each colored point once such that each line is rainbow.
Here's a superficially related notion: the Fano plane, considered as a 3-regular 3-uniform hypergraph, is the smallest such hypergraph that is not 2-colourable. Furthermore, the next smallest is obtained from two copies of Fano by removing an edge in one, a vertex in the other, and gluing appropriately. Are they all similarly obtainable?
@bmreiniger @christianp I found a reference that states there are several constructions for building families of non-2-colourable 3-regular 3-uniform hypergraphs but it doesn't elaborate (or cite a source) https://www.sciencedirect.com/science/article/pii/S0195669813000607
But if I recall correctly, our computer search found only three such hypergraphs up to n=19 all obtained in the above way. On the other hand that construction never includes a 2-cycle (2 vertices sharing 2 edges), which is a strange sufficient condition for a 2-col
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