Generalized polygon

In mathematics, a generalized polygon is an incidence structure introduced by Jacques Tits in 1959. Generalized "n"-gons encompass as special cases projective planes (generalized triangles, "n" = 3) and generalized quadrangles ("n" = 4). Many generalized polygons arise from groups of Lie type, but there are also exotic ones that cannot be obtained in this way. Generalized polygons satisfying a technical condition known as the "Moufang property" have been completely classified by Tits and Weiss. Every generalized "n"-gon with "n" even is also a near polygon.
Definition.
A generalized "2"-gon (or a digon) is an incidence structure with at least 2 points and 2 lines where each point is incident to each line.
For "formula_1" a generalized "n"-gon is an incidence structure (formula_2), where formula_3 is the set of points, formula_4 is the set of lines and formula_5 is the incidence relation, such that:
An equivalent but sometimes simpler way to express these conditions is: consider the bipartite "incidence graph" with the vertex set formula_10 and the edges connecting the incident pairs of points and lines.
From this it should be clear that the incidence graphs of generalized polygons are Moore graphs.
A generalized polygon is of order "(s,t)" if:
We say a generalized polygon is thick if every point (line) is incident with at least three lines (points). All thick generalized polygons have an order.
The dual of a generalized "n"-gon (formula_2), is the incidence structure with notion of points and lines reversed and the incidence relation taken to be the converse relation of formula_14. It can easily be shown that this is again a generalized "n"-gon.
Restriction on parameters.
Walter Feit and Graham Higman proved that "finite" generalized "n"-gons of order ("s", "t") with
"s" ≥ 2, "t" ≥ 2 can exist only for the following values of "n":
Generalized "n"-gons for these values are referred to as generalized digons, triangles, quadrangles, hexagons and octagons.
When Feit-Higman theorem is combined with the Haemers-Roos inequalities, we get the following restrictions,
Every known finite generalized hexagon of order ("s", "t") for "s", "t" > 1 has order 
where "q" is a prime power.
Every known finite generalized octagon of order ("s", "t") for "s", "t" > 1 has order
where "q" is an odd power of 2.
Semi-finite generalized polygons.
If "s" and "t" are both infinite then generalized polygons exist for each "n" greater or equal to 2. It is unknown whether or not there exist generalized polygons with one of the parameters finite (and bigger than "1") while the other infinite (these cases are called "semi-finite"). Peter Cameron proved the non-existence of semi-finite generalized quadrangles with three points on each line, while Andries Brouwer and Bill Kantor independently proved the case of four points on each line. The non-existence result for five points on each line was proved by G. Cherlin using Model Theory. No such results are known without making any further assumptions for generalized hexagons or octagons, even for the smallest case of three points on each line.
Combinatorial applications.
As noted before the incidence graphs of generalized polygons have important properties. For example, every generalized "n"-gon of order "(s,s)" is a "(s+1,2n)" cage. They are also related to expander graphs as they have nice expansion properties. Several classes of extremal expander graphs are obtained from generalized polygons. In Ramsey theory, graphs constructed using generalized polygons give us some of the best known constructive lower bounds on offdiagonal Ramsey numbers.