The non-structure theorems we have discussed in Chapter III rests usually on some freedom on finite sequences and on a kind of order. When our freedom is related to infinite sequences, and to trees, our work is sometimes harder. In particular, we may consider, for λ ≥ χ, χ regular, and φ = φ(x̄0, . . . , x̄α, . . .)α<χ in a vocabulary τ : (∗) For any I ⊆ λ we have a τ–modelMI and sequences āη (fo...