Erdős-Ko-Rado theorems for ovoidal circle geometries and polynomials over finite fields

Stability of Erdős–Ko–Rado theorems in circle geometries

“…We call this incidence structure an extended Laguerre plane . Combinatorially, the most important difference between a Laguerre plane and an extended Laguerre plane, is that in an extended Laguerre plane, two distinct circles intersect in either zero or in two points, but never in exactly 1, see, for example, [1, Section 5.4].…”

“…In some circle geometries, these relations constitute an association scheme. We list these cases, and give the corresponding matrix of eigenvalues, see [1].…”

“…Denote the intersection numbers of these association schemes as $${p}_{ij}^{k}$$. These numbers can be found in [1]. The relevant ones are $$${p}_{33}^{1}=\frac{q}{4}(q-2+\rho )(q-4+\rho ),\unicode{x02007}\unicode{x02007}{p}_{33}^{2}=\frac{q-1}{4}{(q-2+\rho )}^{2}.$$$ To construct $$N$$, assume that the circles of $$L$$ are ordered in $$q+1-\rho $$ blocks of size $$q$$, each representing a parallel class in the affine residue at $$P$$.…”

“…Denote the intersection numbers of these association schemes as p ij k . These numbers can be found in [1]. The relevant ones are…”

“…In [1], the author proved an Erdős–Ko–Rado‐type theorem in ovoidal circle geometries. More specifically, it was shown that if the circle geometry is of order $$q\gt 2$$, and not a Möbius plane of odd order, then the largest intersecting families are exactly the collections of circles through a fixed point, or through a fixed nucleus if the circle geometry is a Laguerre plane of even order.…”