Ups and Downs of First Order Sentences on Random Graphs

whenever is a fixed positive irrational. This raises the question what zero-one valued functions on the positive irrationals arise as the limit probability of a first order sentence on these graphs. Here we prove two necessary conditions on these functions, a number-theoretic and a complexity condition. We hope to prove in a subsequent paper that these conditions together with two simpler and previously proved conditions are also sufficient and thus they constitute a characterization.

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Received October 2, 1998

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Spencer, J., Tardos, G. Ups and Downs of First Order Sentences on Random Graphs. Combinatorica 20, 263–280 (2000).

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  • AMS Subject Classification (1991) Classes:  05C80, 60F20