COMBINATORIAL GEOMETRY

1) Graphs, Hypergraphs, Quasigroups – Main concepts (linear hypergraphs, uniform hypergraphs, transversals, blocking sets – Berge's conjecture for linear hypergraphs.

2) Steiner systems - Hystorical introduction – Necessary existence conditions - Open problems – STS(v) - Bose's construction and Skolem's construction – Systems KTS(v) – Definitions and theorems about the spectrum of S2(2,3,v), S3(2,3,v), S4(2,3,v), S5(2,3,v), S6(2,3,v) – Cyclic STS – Difference method – The problem of Heffter and Peltesohn's theorem – S(2,4,v) – SQS(v) – The problem of the parallel blocks – Costruzions of Steiner STS(v), SQS(v), S(2,4,v), S(2,k,v) – Blocking sets – Berge's conjecture for S(2,k,v).

3) BIBD: definition and hystorical introduction - Fisher's theorem – Symmetric BIBD – Affine and projective planes – Theorem of Bruck-Chowla-Ryser.

4) G-Designs and Hypergraph-designs: G-decompositions – Definitions, theorems - Spectrum for: P3-designs, C4-designs, S4-designs, P4-designs, P5-designs, P6-designs – Case of (K3+e)-designs, (K4-e)-designs – Balanced and strongly balanced G-Designs – Perfect G-designs perfetti - Hypergraph-designs: case of hyperpaths - The method of the difference matrix for the construction of H-designs.

1) C. Berge: "Hypergraphs", North-Holland (1989)

2) C.C.Lindner-C.Rodger: "Design Theory", CRC Boca Raton (2007)

3) M.Gionfriddo, L.MIlazzo, V.Voloshin: "Hypergraphs and Designs", Nova Science Publishers, New York (2015)