Abstract
We consider the random graph \(M^n_{\bar{p}}\) on the set [n], where the probability of {x,y} being an edge is p|x − y|, and \(\bar{p}=(p_1,p_2,p_3,...)\) is a series of probabilities. We consider the set of all \(\bar{q}\) derived from \(\bar{p}\) by inserting 0 probabilities into \(\bar{p}\), or alternatively by decreasing some of the p i . We say that \(\bar{p}\) hereditarily satisfies the 0-1 law if the 0-1 law (for first order logic) holds in \(M^n_{\bar{q}}\) for every \(\bar{q}\) derived from \(\bar{p}\) in the relevant way described above. We give a necessary and sufficient condition on \(\bar{p}\) for it to hereditarily satisfy the 0-1 law.
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Shelah, S., Doron, M. (2010). Hereditary Zero-One Laws for Graphs. In: Blass, A., Dershowitz, N., Reisig, W. (eds) Fields of Logic and Computation. Lecture Notes in Computer Science, vol 6300. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15025-8_29
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DOI: https://doi.org/10.1007/978-3-642-15025-8_29
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15024-1
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