TY - JOUR

T1 - On Betti numbers of flag complexes with forbidden induced subgraphs

AU - Adiprasito, Karim

AU - Nevo, Eran

AU - Tancer, Martin

N1 - 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.

PY - 2020/1/1

Y1 - 2020/1/1

N2 - 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.

AB - 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.

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

UR - http://www.scopus.com/inward/record.url?scp=85067056342&partnerID=8YFLogxK

U2 - 10.1017/S030500411900001X

DO - 10.1017/S030500411900001X

M3 - Article

AN - SCOPUS:85067056342

VL - 168

SP - 567

EP - 600

JO - Mathematical Proceedings of the Cambridge Philosophical Society

JF - Mathematical Proceedings of the Cambridge Philosophical Society

SN - 0305-0041

IS - 3

ER -