Literature:
[H] M. Hall: Combinatorial Theory, Wiley 1986
[BR] A. Beutelspacher, U. Rosenbaum: Projective Geometry: From Foundations to Applications, Cambridge University Press 1998
Requests for the exam All handouts can be found in SIS.
Feb 20, 2025 - Finite projective planes, orthogonal Latin squares
Axiomatic definition of finite projective plane, connection between finite
projective planes and orthogonal Latin squares
useful handout from NMAG403-Combinatorics course
[H] chapters 13.1, 12.3, 13.2
Feb 27, 2025 - Finite affine planes, intro to Block designs
Axiomatic definition of finite affine planes, connection between finite
projective and finite affine planes
handout (in Czech)
[BR] Theorem 1.6.4
Intro to Balanced Incomplete Block Designs, divisibility conditions, Fisher's inequality
useful handout from NMAG403-Combinatorics course
[H] chapters 10.1., 10.2
Mar 6, 2025 --- There is no class --- !!
Suggested reading: symmetric block designs, Bruck-Ryser-Chowla theorem
useful handout from NMAG403-Combinatorics course
[H] chapter 10.3
Mar 13, 2025 - Not-so-regular Block designs, Mutually Orthogonal Latin Squares
Block designs with blocks of different sizes, transaprent sets of blocks, connection to mutually orthogonal Latin squares. Resolvable designs, group divisible
designs. MSOL(4t+2)>=2 for every t>=2.
Mar 20, 2025 - Finite projective planes
Thm: If FPP(n) exists for n=(1 or 2)mod 4, then n=a^2+b^2.
Mar 27, 2025 - Finite projective geometry I
Axiomatic definition of FPG, Singer's construction. Relation to BIBDs.
Apr 3, 2025 - Finite projective geometry II
Basic properties, Steintz exchange theorem, deinition of dimension.
Apr 10, 2025 - Finite projective geometry III
Dimension of intersection and union of subspaces. Subspaces of dimension 2 are finite projective planes.
Apr 17, 2025 - Finite projective geometry IV
Isomorphisms of set systems. Automorphisms of FPG are collinearity preserving bijections. Any two planes in a FPG are isomorphic. Hyperplanes in FPG form a symmetric BIBD.
Apr 24, 2025 - Finite projective geometry V
FPG of dimension > 2 is Desarguesian. Indtroduction to collineations.
May 15, 2025 - Finite projective geometry VI
Central collineations. A collineation of P-line can be extended to a collineation of the entire space. A central collineation is uniquely determined by the image of a single point which is not its center and does not lie in its axis.
May 22, 2025 - Finite projective geometry VII
Baer theorem. Calculus with collineations. Veblen-Young theorem (the weak version) - if FPG(q) exists and has dimension >2, then q (the order of the geometry) is a power of a prime.