An intersection theorem for systems of sets

Erdos and Rado defined a Δ-system, as a family in which every two members have the same intersection. Here we obtain a new upper bound on the maximum cardinality φ(n, q) of an n-uniform family not containing any Δ-system of cardinality q. Namely, we prove that, for any α > 1 and q, there exists C = C(α, q) such that, for any n, φ(n, q) ≤ Cn!((log log log n)²/α log log n)ⁿ.