COMBINATORIAL GEOMETRY

The main goal is to introduce the student to the study of Graphs, Hypergraphs and to know the main tecniques of combinatorial theories. Mainy open problems are pointed out.

The course is organized in frontal lessons and research-seminars. Pratic exercises are considered. Open problem will be examined.

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.

5) H-designs: H(3)-designs, H(4)-designs, constructions and spectrum - The case of P(h)(h-1,h+1)-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)