An Inequality on the Size of a Set in a Cartesian Product

Let Ω be a finite subset of the Cartesian productW1 × ⋯ × Wnof n sets. ForA ⊂ {1, 2,⋯ , n }, denote by ΩAthe projection ofΩ onto the Cartesian product of Wi, i ∈ A. Generalizing an inequality given in an article by Shen, we prove that | Ω |2 ≤ |ΩA1 |⋯| ΩAk| provided that { A1,⋯ , Ak} is a double cover of {1, 2,⋯ , n }. This inequality is applied to give some bounds on the numbers of special subgraphs of a graph.