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Abstract: Classical results by Behrend, Erdös, Pillai, Szemerédi, the authors, and others are improved and extended to other concepts of densities in two directions, based on smooth and multiplicative weightings. [Copyright &y& Elsevier]

Abstract: In Ahlswede et al. [Discrete Math. 273(1–3) (2003) 9–21] we posed a series of extremal (set system) problems under dimension constraints. In the present paper, we study one of them: the intersection problem. The geometrical formulation of our problem is as follows. Given integers , determine or estimate the maximum number of -vectors in a k-dimensional subspace of the Euclidean n-space , such that the inner product (“intersection”) of any two is at least t. Also we are interested in the restricted (or the uniform) case of the problem; namely, the problem considered for the -vectors of the same weight . The paper consists of two parts, which concern similar questions but are essentially independent with respect to the methods used. In Part I, we consider the unrestricted case of the problem. Surprisingly, in this case the problem can be reduced to a weighted version of the intersection problem for systems of finite sets. A general conjecture for this problem is proved for the cases mentioned in Ahlswede et al. [Discrete Math. 273(1–3) (2003) 9–21]. We also consider a diametric problem under dimension constraint. In Part II, we study the restricted case and solve the problem for and , and also for any fixed and k large. [Copyright &y& Elsevier]

We discovered a new and simple shifting technique. It makes it possible to prove results on shadows like the Kruskal–Katona theorem without any additional arguments.As another application we obtain the following new result. For s, d, k∈N, 1≤d≤s, d≤k define the subclass of Nk (the k-subsets of N) B(k,s,d)=B∈Nk:|B∩[1,s]|≥d. Let A⊂B(k,s,d) and |A|=m. Then the cardinality of the ℓ-shadow of A is minimal if A consists of the first m elements of B(k,s,d) in colexicographic order. A more general form of this result is given as well. Other applications are to be expected. [Copyright &y& Elsevier]

Let I(n, t) be the class of all t -intersecting families of subsets of [ n ] and set Ik(n, t) = I (n, t) ∩ 2[ n ]k, I ≤ k(n, t) = I(n, t) ∩ 2[ n ] ≤ k.After the maximal families inI (n, t) [13] and in Ik(n, t) [1,9] are known we study now maximal families in I ≤ k(n, t). We present a conjecture about the maximal cardinalities and prove it in several cases.More generally cardinalities are replaced by weights and asymptotic estimates are given.Analogous investigations are made for I(n,t ) ∩ C(n, s), where C(n,s ) is the class of all s -cointersecting families of subsets of [ n ]. In particular we establish an asymptotic form of a conjecture by Bang et al. [4]. [Copyright &y& Elsevier]