After our summer break our regular study group resumed on Friday. This week, John started what looks like being a two week discussion about and generalised hexagons.

A *generalised polygon (n-gon)* of order *(s,t) *is a *1-(v,s+1,t+1) *design (that is, a set of *v *points and a collection of *(s+1)*-subsets called blocks, or lines, such that each point lies in *t+1 *blocks) whose incidence graph has girth *2n *and diameter *n*. A polygon with *n* vertices and the edges taken to be the blocks, is a generalised *n*-gon with *s=t=1*. The incidence graph of such a polygon is a cycle of length *2n*. We call a generalised polygon *thick *if .

Some authors define a generalised *n*-gon to be a bipartite graph with girth *2n* and diameter *n*. John does not like this definition as it is referring to the incidence graph and not the design/geometry. This reminds me of a question Alice, Gordon and I had a couple of weeks ago which we were not able to resolve. We were wondering if “bipartite” was necessary in this definition. That is, is a graph of girth *2n* and diameter *n* bipartite? Any answers are more than welcome.

The incidence graph of a generalised 2-gon is a complete bipartite graph. The generalised 3-gons are the projective planes and the generalised 4-gons, or generalised quadrangles, are the rank two polar spaces.

Walter Feit and Graham Higman proved in 1964 (in the first volume of the Journal of Algebra) that finite thick generalised n-gons only exist for and . While there are many projective planes and generalised quadrangles, there are only two known infinite families of generalised hexagons and one known infinite family of generalised octagons.

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