Abstract To a large extent the present work is far from being conclusive, instead, new directions of research in combinatorial extremal theory are started. Also questions concerning generalizations are immediately noticeable. The insentive came from problems in several fields such as Algebra, Geometry, Probability, Information and Complexity Theory. Like several basic combinatorial problems they may play a role in other fields. For scenarious of interplay we refer also to [R. Ahlswede, Advances on extremal problems in number theory and combinatorics, European Congress of Mathematics, Barcelona 2000, Vol. I, 147–175, Carles Casacuberta, Rosa Maria Miro–Roig, Joan Verdera, Sebastia Xambo–Descamps, edit., Progress in Mathematics, Vol. 201, Birkhauser Verlag, Basel–Boston–Berlin, 2001].