### Abstract

Shelah and Spencer [1] proved that the zero-one law holds for the first order sentences on random graphs G(n,n^{-α}) 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) |
---|---|

Pages (from-to) | 263-280 |

Number of pages | 18 |

Journal | Combinatorica |

Volume | 20 |

Issue number | 2 |

State | Published - 2000 |

### Fingerprint

### ASJC Scopus subject areas

- Mathematics(all)
- Discrete Mathematics and Combinatorics

### Cite this

*Combinatorica*,

*20*(2), 263-280.

**Ups and downs of first order sentences on random graphs.** / Spencer, Joel; Tardos, Gábor.

Research output: Contribution to journal › Article

*Combinatorica*, vol. 20, no. 2, pp. 263-280.

}

TY - JOUR

T1 - Ups and downs of first order sentences on random graphs

AU - Spencer, Joel

AU - Tardos, Gábor

PY - 2000

Y1 - 2000

N2 - Shelah and Spencer [1] proved that the zero-one law holds for the first order sentences on random graphs G(n,n-α) 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.

AB - Shelah and Spencer [1] proved that the zero-one law holds for the first order sentences on random graphs G(n,n-α) 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.

UR - http://www.scopus.com/inward/record.url?scp=0034355079&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0034355079&partnerID=8YFLogxK

M3 - Article

VL - 20

SP - 263

EP - 280

JO - Combinatorica

JF - Combinatorica

SN - 0209-9683

IS - 2

ER -