10 MIRNA DZAMONJAˇ length ω3. We first describe the iteration. It will have two kinds of coordinates. In each coordinate of the first kind. of it we are given a name for a petition Γ ˜ on the approximation family and we force to embed MΓ ˜ into MΓ∗ ˜ for some quorumed petition Γ∗. ˜ By remarks above, for our final universality result it will be suﬃcient to ensure that there are ℵ2 triangle-free graphs on ω1 which embed all MΓ∗ for quorumed petitions Γ∗. By bookkeeping, we shall at each stage α ω2 of the main iteration assure by Qα ˜ of forcing that we have dealt with all quorumed petitions in V Pα . In fact we shall assure that there is in V Pα+1 a single triangle-free graph Gα ∗ on ω1 which embeds all MΓ∗ for quorumed petitions Γ∗ in V Pα . This is where the preliminary forcing of the block α comes in: in it we introduce a system Sα of members of the approximation family indexed by the nodes in T in a such a way that each branch through T gives a petition in the approximation family. It is easy to see that the union of this system is a triangle-free graph on ω1, which will be our G∗ α . In the second kind of coordinates in the block α we shall be embedding a quorumed petition H ˜ given by the bookkeeping into the subsystem of Sα given by the elements indexed by the nodes on some branch of T . This assures that MH embeds into G∗ α . The main point of the proof is to make sure that each individual forcing in a block is ccc, and in assuring so in the second kind of coordinates we get to use an amalgamation property possessed by the elements of the approximation family K: Suppose that M0, N0, M1, M2, N1, N2 and M are in K such that • M0 = M1 ∩ M2 and M1 is isomorphic with M2 by an isomorphism which is identity on M0, • N0 = N1 ∩ N2 and N1 is isomorphic with N2 by an isomorphism which is identity on N0, • each Mi is an induced subgraph of the corresponding Ni and the universe of Ml consists of even ordinals in the universe of Nl, • there are limit ordinals δ0 δ1 δ2 such that Nl ⊆ δl for each l 3, • the universe of M is contained in the even ordinals and M1, M2 are induced subgraphs of M. Then there is N ∈ K whose induced subgraphs include M, N1 and N2. This property is called workability in [3]. Notice that checking that it is true really uses the definition of the approximation family, not only the properties of the class of triangle-free graphs. The proof in the case λ+ for λ = λλ in place of ℵ1 has to deal with a strong version of λ+-cc needed in order to iterate, which introduces additional complica- tions which we shall not describe here. The structure of the proof as described above uses a tree of models rather than a linearly organised structure like in [9]. The price we have to pay is that the final universe does not have one universal model, rather just a small universal family. The following question is still open: Question 3.6. Is it consistent to have a universal triangle-free graph on ℵ1 and not CH? 3.3. Linear orders. The next step to consider in our increasing level of com- plexity of theories is a theory that does not have a workable approximation family. Such a theory is the theory of dense linear orders. Namely, the main result of Kojman-Shelah [7] is that this theory is highly non-amenable, and the proof we presented for triangle-free graphs cannot be adopted to this case- as it would show

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