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Mathematics > Combinatorics

arXiv:1809.08333 (math)
[Submitted on 21 Sep 2018 (v1), last revised 3 Jul 2019 (this version, v2)]

Title:Evolving Shelah-Spencer Graphs

Authors:Richard Elwes
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Abstract:An \emph{evolving Shelah-Spencer process} is one by which a random graph grows, with at each time $\tau \in {\bf N}$ a new node incorporated and attached to each previous node with probability $\tau^{-\alpha}$, where $\alpha \in (0,1) \setminus {\bf Q}$ 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 [Spencer, J., 2013. The strange logic of random graphs (Vol. 22). Springer Science & Business Media.] and throughout the model-theoretic literature. The first order axiomatisation for classical Shelah-Spencer graphs comprises a 'Generic Extension' axiom and a 'No Dense Subgraphs' axiom. We show that in our context 'Generic Extension' continues to hold. While 'No Dense Subgraphs' fails, a weaker 'Few Rigid Subgraphs' property holds.
Subjects: Combinatorics (math.CO)
MSC classes: 05C80
Cite as: arXiv:1809.08333 [math.CO]
  (or arXiv:1809.08333v2 [math.CO] for this version)
  https://doi.org/10.48550/arXiv.1809.08333

Submission history

From: Richard Elwes [view email]
[v1] Fri, 21 Sep 2018 22:22:53 UTC (13 KB)
[v2] Wed, 3 Jul 2019 22:04:57 UTC (16 KB)
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