Definition 19.1. 1. The mapping F : X → P(X) on the set X is said to be a set mapping if x ∉ F(x) for each x ∈ X.

2. We say that the above set mapping has order λ if |F(x)| < λ for each x ∈ X.

3. We call the set S ⊂ X free with respect to F if x ∉ F(y) for each x,y ∈ S.

We observe that, given a set mapping F of order λ on X, if Y ⊂ X and Fy (X) = Y ∩ F(X), then the set mapping is also of order λ, and every set S ⊂ Y that is free with respect to Fy is also free with respect to F. We will frequently use this observation without calling attention to it explicitly.

It was P. Turán who first pointed out that if the order of F is small and X is large, then often there is a large free subset. He proved the existence of a free subset of cardinality 2ℵ0 in the case when λ = ω and |X| = 2ℵ0. The next lemma was established by D. Lázár in 1936.

Lemma 19.1.Assume K ≥ ω is regular, K > λ, and F is a set mapping of order λ on k. Then there is a set S ⊂ k of cardinality k that is free with respect to F.