Abstract
This thesis focuses on the interplay of extremal combinatorics and finite geometry. Combinatorics is concerned with discrete (and usually finite) objects. Extremal combinatorics studies how large or how small a collection of finite objects can be under certain restrictions. Those objects can be sets, graphs, vectors, etc. These questions are often motivated by problems in information theory and computer science. Another branch of combinatorics is finite geometry over finite fields of order q. Although there is no field
of order 1, certain substructures in finite geometry can be interpreted as versions of projective spaces for q = 1. A triangle in a projective plane is a good example of this phenomenon.
Following an introductory chapter, Chapter 2 gives an overview of some classical problems and their qanalogues in extremal combinatorics. The most recent results for the qanalogues of Sperner's theorem, the ErdösKoRado theorem and several versions of Bollobás's theorem are described. For the qanalogue of the HiltonMilner theorem we give some new results. We also give a new bound for the minimum size of the qanalogue of small maximal cliques. We conclude that sometimes results for a qanalogue can be obtained by using the same technique as in the original problem. In some cases the answer to the problem is even identical to that of the original problem. In other cases techniques used for the qanalogue could be used for improving bounds for the original problem.
The third chapter describes the known results for the chromatic number of the Kneser graphs and gives new bounds for the chromatic number of the qanalogue. The vertices of a Kneser graph are subsets (of a fixed size) of a set, whereas two vertices are adjacent if they are disjoint in their subset representation. In 1955 M. Kneser conjectured the chromatic number of those graphs. In 1978 this was proven correct by L. Lóvasz. Two small cases in the projective space qanalogue were solved in 2001 by J. Eisfeld, L. Storme and P. Sziklai and in 2006 by A. Chowdhury, C. Godsil and G. Royle. We
describe an asymptotic result for all cases (except for one parameter family, where we give a partial proof) using the bounds from the qanalogue of the HiltonMilner theorem.
Chapters 4, 5 and 6 describe other qanalogues of the Kneser graphs.
In Chapter 4 we define a family of Kneser type graphs over pairs of incident points and hyperplanes of a projective space. We describe large maximal cocliques in these graphs and prove that for small dimensions these are the largest possible cocliques. We conjecture that this is the case for all dimensions.
In Chapter 5 we extend the Kneser graphs to the case of finite polar spaces instead of finite projective spaces and in some cases give descriptions and chromatic numbers of these graphs.
Chapter 6 describes the generalization of Kneser graphs over coset spaces of Chevalley groups (parameterized by q) with respect to parabolic subgroups. This encompasses all previous cases and extends to some interesting new cases. First we describe these
graphs when q = 1 and give chromatic numbers for some families. Then we consider the graphs defined over coset spaces of Chevalley groups for general q with respect to parabolic subgroups and study what results of the q = 1 case can be translated to the general case. In this thesis we found different possibilities for the connection between
the bounds for the q = 1 case and the case for general q. In some cases the bounds for the general case are identical to the bounds for q = 1. In other cases we obtain the bounds for q = 1 by taking the limit for q¿1 in the general bound. Other cases show a completely different bound in both cases.
Original language  English 

Qualification  Doctor of Philosophy 
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Award date  16 Apr 2009 
Place of Publication  Eindhoven 
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Publication status  Published  2009 