# Ups and downs of first order sentences on random graphs

Joel Spencer, Gábor Tardos

Research output: Contribution to journalArticle

### Abstract

Shelah and Spencer  proved that the zero-one law holds for the first order sentences on random graphs G(n,n) whenever α 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.

Original language English (US) 263-280 18 Combinatorica 20 2 Published - 2000

### Fingerprint

Random Graphs
First-order
Zero-one Law
Sufficient
Necessary Conditions
Zero
Graph in graph theory

### ASJC Scopus subject areas

• Mathematics(all)
• Discrete Mathematics and Combinatorics

### Cite this

In: Combinatorica, Vol. 20, No. 2, 2000, p. 263-280.

Research output: Contribution to journalArticle

Spencer, J & Tardos, G 2000, 'Ups and downs of first order sentences on random graphs', Combinatorica, vol. 20, no. 2, pp. 263-280.
Spencer, Joel ; Tardos, Gábor. / Ups and downs of first order sentences on random graphs. In: Combinatorica. 2000 ; Vol. 20, No. 2. pp. 263-280.
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