Extremal generalized hexagons and triple intersection numbers

Abstract

A triple intersection number in a finite generalized hexagon S of order (s,t) is a number of the form |Γi(x)∩Γj(y)∩Γk(ω)|, where x, y are two points and ω is either a point z or a line L. The fact that S is extremal, i.e. satisfies t=s3>1, implies that certain combinatorial properties regarding triple intersection numbers hold. The earliest result in this direction is due to Haemers who showed that the number |Γi(x)∩Γj(y)∩Γk(L)| only depends on s, i, j, k and the configuration determined by (x,y,L). In this paper, we prove a similar result for the triple intersection numbers that involve three points, with exception of the case where the points are mutually opposite. In this case, we show that the 64 triple intersection numbers only depend on s and one extra parameter. We also show how Haemers’ original results can be deduced from the results obtained here, leading to a more elementary entirely combinatorial treatment where the use of any eigenvalue techniques and other non-elementary matrix theory are avoided.