discipline

Pure Math

subject

Symmetric combinatorial structures

lecturers

Tamás Szőnyi, Péter Sziklai, András Gács

credits

2

period

1, 3

curriculum

Projective and affine planes, Latin squares.

Planes over a field and their representations. 

t-(v,k,l) designs. Restrictions on the parameters.  Fisher's inequalty  (for t=2).

Skew intersecting systems, square designs. Linear algebra methods.

Some more non-existence results: the Bruck-Ryser-Chowla theorem and its variants. Problems concerning the existence of projective planes.

Characterization of the design of points and hyperplanes of a projective space (Dembowski-Wagner); the parameters do not characterize the design (Kantor).

Biplanes (lambda=2), Hadamard matrices and designs  (t=2). Some examples, Paley design, PG(n-1,2).

Strongly regular graphs: restrictions on the parameters. The point-graph of designs and strongly regular graphs. “Switching” of strongly regular graphs (Seidel). Examples: ladder graph and triangular graph, etc.

Recursive construction of designs: point- and blockresidual design, extension. Hadamard 3-designs, extension of affine planes (inversive planes). Extension of projective planes and symmetric designs in general (Cameron's theorem). Sketch of construction of the Witt designs for the Matieu groups. Asymptotic results for the existence of designs  (Wilson's theorem, Teirlinck's theorem for the existence of designs with t>5 without proof). Difference sets and Hall's multiplyer theorem.

 

 

literature

Cameron-Van Lint: Designs, graphs, codes and their links, Cambridge University Press, Cambridge, 1991.

form of tuition

lectures

mode of assessment

Oral exam