We consider the conjecture stating that a matrix with rank o(n) and ones on the main diagonal must contain nonzero entries on a 2 \Theta 2 submatrix with one entry on the main diagonal.

We give a generalization for the Deza-Frankl-Singhi Theorem in case of multiple intersections. More exactly, we prove, that if H is a set-system, which satisfies that for some k, the k-wise intersections occupy only ` residue-classes modulo a p prime, while the sizes of the members of H are not in these residue classes, then the size of H is at most (k \Gamma 1) ` X i=0 / n i ! This result considerably strengthens an upper bound of Furedi (1983), and gives partial answer to a question of T. S'os (1976). As an application, we give explicit constructions for coloring the k-subsets of an n element set with t colors, such that no monochromatic complete hypergraph on exp(c k (log n log log n) 1=t ) vertices exists. By our best knowledge, this is the first explicit construction of a Ramsey-hypergraph.

We have determined all exponents of (n; k; L)-systems of k 12 except for essentially two cases, which are related to the Steiner systems S(11; 5; 4) and S(12; 6; 5). This requires several new constructions. Also some refinements of the previous methods are necessary to get suitable upper bounds. 1

We prove a version of the Ray-Chaudhuri-Wilson and Frankl-Wilson theorems for k- wise intersections and also generalize a classical code-theoretic result of Delsarte for k-wise Hamming distances. A set of code-words a 1 ; a 2 ; : : : ; a k of length n have k-wise Hamming-distance `, if there are exactly ` such coordinates, where not all of their coordinates coincide (alternatively, exactly n ` of their coordinates are the same). We show a Delsarte-like upper bound: codes with few k-wise Hamming-distances must contain few code-words.