TY - JOUR

T1 - On subspaces contained in subsets of finite homogeneous spaces

AU - Karpovsky, M. G.

AU - Milman, V. D.

PY - 1978

Y1 - 1978

N2 - 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, qk-1) of a subset M in a linear n-space Enq over GE(q) such that M does not contain any k-subspace of Enq. We shall consider also some applications of this result.

AB - 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, qk-1) of a subset M in a linear n-space Enq over GE(q) such that M does not contain any k-subspace of Enq. We shall consider also some applications of this result.

UR - http://www.scopus.com/inward/record.url?scp=4243187875&partnerID=8YFLogxK

U2 - 10.1016/0012-365X(78)90060-2

DO - 10.1016/0012-365X(78)90060-2

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AN - SCOPUS:4243187875

SN - 0012-365X

VL - 22

SP - 273

EP - 280

JO - Discrete Mathematics

JF - Discrete Mathematics

IS - 3

ER -