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A counterexample to the Bollobás–Riordan conjectures on sparse graph limits

Part of: Extremal combinatorics Graph theory

Published online by Cambridge University Press:  27 January 2021

Ashwin Sah ,
Mehtaab Sawhney ,
Jonathan Tidor  and
Yufei Zhao
Ashwin Sah
Affiliation:
Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Mehtaab Sawhney
Affiliation:
Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Jonathan Tidor
Affiliation:
Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Yufei Zhao*
Affiliation:
Massachusetts Institute of Technology, Cambridge, MA 02139, USA
*
*Corresponding author. Email: [email protected]
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Abstract

Bollobás and Riordan, in their paper ‘Metrics for sparse graphs’, proposed a number of provocative conjectures extending central results of quasirandom graphs and graph limits to sparse graphs. We refute these conjectures by exhibiting a sequence of graphs with convergent normalized subgraph densities (and pseudorandom C4-counts), but with no limit expressible as a kernel.

MSC classification

Primary: 05C35: Extremal problems
Secondary: 05D99: None of the above, but in this section
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 30 , Issue 5 , September 2021 , pp. 796 - 799
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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Footnotes

†

Supported by NSF Graduate Research Fellowship Program DGE-1122374.

‡

Supported by NSF Award DMS-1764176, the MIT Solomon Buchsbaum Fund, and a Sloan Research Fellowship.

References

Bollobás, B. and Riordan, O. (2009) Metrics for sparse graphs. In Surveys in Combinatorics 2009, Vol. 365 of London Mathematical Society Lecture Note Series, pp. 211–287. Cambridge University Press.Google Scholar
Borgs, C., Chayes, J. T., Cohn, H. and Zhao, Y. (2018) An Lp theory of sparse graph convergence, II: LD convergence, quotients and right convergence. Ann. Probab. 46 337–396.10.1214/17-AOP1187CrossRefGoogle Scholar
Borgs, C., Chayes, J. T., Cohn, H. and Zhao, Y. (2019) An Lp theory of sparse graph convergence, I: limits, sparse random graph models, and power law distributions. Trans. Amer. Math. Soc. 372 3019–3062.Google Scholar
Borgs, C., Chayes, J. T., Lovász, L., Sós, V. T. and Vesztergombi, K. (2008) Convergent sequences of dense graphs, I: subgraph frequencies, metric properties and testing. Adv. Math. 219 1801–1851.CrossRefGoogle Scholar
Chung, F. R. K., Graham, R. L. and Wilson, R. M. (1989) Quasi-random graphs. Combinatorica 9 345–362.CrossRefGoogle Scholar
Conlon, D., Fox, J. and Zhao, Y. (2014) Extremal results in sparse pseudorandom graphs. Adv. Math. 256 206–290.CrossRefGoogle Scholar
Conlon, D., Fox, J. and Zhao, Y. (2014) The Green–Tao theorem: an exposition. EMS Surv. Math. Sci. 1 249–282.10.4171/EMSS/6CrossRefGoogle Scholar
Conlon, D., Fox, J. and Zhao, Y. (2015) A relative Szemerédi theorem. Geom. Funct. Anal. 25 733–762.CrossRefGoogle Scholar
Green, B. and Tao, T. (2008) The primes contain arbitrarily long arithmetic progressions. Ann. of Math. (2) 167 481–547.10.4007/annals.2008.167.481CrossRefGoogle Scholar
Kohayakawa, Y. (1997) Szemerédi’s regularity lemma for sparse graphs. In Foundations of Computational Mathematics (Rio de Janeiro, 1997) (Cucker, F. and Shub, M., eds), pp. 216–230. Springer.Google Scholar
Lovász, L. (2012) Large Networks and Graph Limits, Vol. 60 of American Mathematical Society Colloquium Publications. American Mathematical Society.Google Scholar
Thomason, A. (1987) Pseudorandom graphs. In Random Graphs ’85 (Poznań, 1985), Vol. 144 of North-Holland Mathematics Studies, pp. 307–331. North-Holland.Google Scholar
Thomason, A. (1987) Random graphs, strongly regular graphs and pseudorandom graphs. In Surveys in Combinatorics 1987 (New Cross, 1987), Vol. 123 of London Mathematical Society Lecture Note Series, pp. 173–195. Cambridge University Press.Google Scholar