On subspaces contained in subsets of finite homogeneous spaces

Let E(l) ⊂ E(l + 1) ⊂ · · · ⊂ E(n) be a system of finite sets and H(l) ⊂ H(l + 1) ⊂ · · · ⊂ H(n) be a system of groups, where H(k) is a transitive group of automorphisms of E(k). Denote G(k, E(n)) = {X ⊂ E(n): ∃h ∈ H(n), h(E(k)) = X}. We investigate the following problem: given n, 1 ≤ l ≤ k ≤ n, 0 < λ ≤ |E(k)|. What is the maximal cardinality L(n, k, λ) of a set M ⊆ E(n) such that for ∀X ∈ G(k, E(n)), |X ∩ M| < λ? We shall establish an upper bound for L(n, k, λ) and prove that for some important cases it will coincide with the lower bound for L(n, k, λ). We shall consider the three special cases of our problem: linear spaces, Grassman spaces, Turan's problem. For linear spaces, we obtain the exact formula for the maximal cardinality L(n, k, q^k-1) of a subset M in a linear n-space Eⁿ_q over GE(q) such that M does not contain any k-subspace of Eⁿ_q. We shall consider also some applications of this result.