Abstract
AbstractWe define an evolving Shelah‐Spencer process as one by which a random graph grows, with at each time a new node incorporated and attached to each previous node with probability , where is fixed. We analyse the graphs that result from this process, including the infinite limit, in comparison to Shelah‐Spencer sparse random graphs discussed in [21] and throughout the model‐theoretic literature. The first order axiomatisation for classical Shelah‐Spencer graphs comprises a Generic Extension axiom scheme and a No Dense Subgraphs axiom scheme. We show that in our context Generic Extension continues to hold. While No Dense Subgraphs fails, a weaker Few Rigid Subgraphs property holds.
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