## Abstract

We analyse the asymptotic extremal growth rate of the Betti numbers of clique complexes of graphs on n vertices not containing a fixed forbidden induced subgraph H. In particular, we prove a theorem of the alternative: for any H the growth rate achieves exactly one of five possible exponentials, that is, independent of the field of coefficients, the nth root of the maximal total Betti number over n-vertex graphs with no induced copy of H has a limit, as n tends to infinity, and, ranging over all H, exactly five different limits are attained. For the interesting case where H is the 4-cycle, the above limit is 1, and we prove a superpolynomial upper bound.

Original language | American English |
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Pages (from-to) | 567-600 |

Number of pages | 34 |

Journal | Mathematical Proceedings of the Cambridge Philosophical Society |

Volume | 168 |

Issue number | 3 |

DOIs | |

State | Published - 2020 |

### Bibliographical note

Funding Information:† Supported by ERC-2016-STG 716424 - CASe and Israel Science Foundation grant 1050/16. ‡ Partially supported by Israel Science Foundation grants ISF-805/11, ISF-1695/15, by grant 2528/16 of the ISF-NRF Singapore joint research program, and by ISF-BSF joint grant 2016288. § Partially supported by the GACˇ R grant 16-01602Y and by Charles University project UNCE/SCI/004. Part of this work was done when M. T. was affiliated with IST Austria.

Publisher Copyright:

Copyright © Cambridge Philosophical Society 2019.

## Keywords

- 2010 Mathematics Subject Classification: 05C35 05E45 57M15