## Abstract

We prove several relations on the f-vectors and Betti numbers of flag complexes. For every flag complex Δ, we show that there exists a balanced complex with the same f-vector as Δ, and whose top-dimensional Betti number is at least that of Δ, thereby extending a theorem of Frohmader by additionally taking homology into consideration. We obtain upper bounds on the top-dimensional Betti number of Δ in terms of its face numbers. We also give a quantitative refinement of a theorem of Meshulam by establishing lower bounds on the f-vector of Δ, in terms of the top-dimensional Betti number of Δ. This result has a continuous analog: If Δ is a (d−1)-dimensional flag complex whose (d−1)-th reduced homology group has dimension a≥0 (over some field), then the f-polynomial of Δ satisfies the coefficient-wise inequality f_{Δ}(x)≥(1+(ad+1)x)^{d}.

Original language | American English |
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Article number | 105466 |

Pages (from-to) | 1-21 |

Number of pages | 21 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 182 |

DOIs | |

State | Published - Aug 2021 |

### Bibliographical note

Publisher Copyright:© 2021 Elsevier Inc.

## Keywords

- Betti number
- Clique complex
- Flag complex
- Homology
- Turán graph
- f-vector