A proof of the Kahn–Kalai conjecture

Park, Jinyoung; Pham, Huy

doi:10.1090/jams/1028

A proof of the Kahn–Kalai conjecture
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by Jinyoung Park and Huy Tuan Pham;

J. Amer. Math. Soc. 37 (2024), 235-243

DOI: https://doi.org/10.1090/jams/1028

Published electronically: August 7, 2023

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Abstract:

Proving the “expectation-threshold” conjecture of Kahn and Kalai [Combin. Probab. Comput. 16 (2007), pp. 495–502], we show that for any increasing property $\mathcal {F}$ on a finite set $X$, \[ p_c(\mathcal {F})=O(q(\mathcal {F})\log \ell (\mathcal {F})), \] where $p_c(\mathcal {F})$ and $q(\mathcal {F})$ are the threshold and “expectation threshold” of $\mathcal {F}$, and $\ell (\mathcal {F})$ is the maximum of $2$ and the maximum size of a minimal member of $\mathcal {F}$.

References

Ryan Alweiss, Shachar Lovett, Kewen Wu, and Jiapeng Zhang, Improved bounds for the sunflower lemma, Ann. of Math. (2) 194 (2021), no. 3, 795–815. MR 4334977, DOI 10.4007/annals.2021.194.3.5
T. Bell and A. Frieze, Rainbow powers of a Hamilton cycle in $G_{n,p}$, Preprint, arXiv:2210.08534.
Béla Bollobás, Random graphs, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 73, Cambridge University Press, Cambridge, 2001. MR 1864966, DOI 10.1017/CBO9780511814068
B. Bollobás and A. Thomason, Threshold functions, Combinatorica 7 (1987), no. 1, 35–38. MR 905149, DOI 10.1007/BF02579198
B. DeMarco and J. Kahn, Note on a problem of M. Talagrand, Random Structures Algorithms 47 (2015), no. 4, 663–668. MR 3418910, DOI 10.1002/rsa.20559
P. Erdős and A. Rényi, On random graphs. I, Publ. Math. Debrecen 6 (1959), 290–297. MR 120167, DOI 10.5486/pmd.1959.6.3-4.12
P. Erdős and A. Rényi, On the evolution of random graphs, Magyar Tud. Akad. Mat. Kutató Int. Közl. 5 (1960), 17–61 (English, with Russian summary). MR 125031
Keith Frankston, Jeff Kahn, Bhargav Narayanan, and Jinyoung Park, Thresholds versus fractional expectation-thresholds, Ann. of Math. (2) 194 (2021), no. 2, 475–495. MR 4298747, DOI 10.4007/annals.2021.194.2.2
Keith Frankston, Jeff Kahn, and Jinyoung Park, On a problem of M. Talagrand, Random Structures Algorithms 61 (2022), no. 4, 710–723. MR 4504841, DOI 10.1002/rsa.21077
Ehud Friedgut, Sharp thresholds of graph properties, and the $k$-sat problem, J. Amer. Math. Soc. 12 (1999), no. 4, 1017–1054. With an appendix by Jean Bourgain. MR 1678031, DOI 10.1090/S0894-0347-99-00305-7
Ehud Friedgut, Hunting for sharp thresholds, Random Structures Algorithms 26 (2005), no. 1-2, 37–51. MR 2116574, DOI 10.1002/rsa.20042
Geoffrey Grimmett, The random-cluster model, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 333, Springer-Verlag, Berlin, 2006. MR 2243761, DOI 10.1007/978-3-540-32891-9
Johan Torkel Hastad, COMPUTATIONAL LIMITATIONS OF SMALL DEPTH CIRCUITS, ProQuest LLC, Ann Arbor, MI, 1986. Thesis (Ph.D.)–Massachusetts Institute of Technology. MR 2941107
Svante Janson, Tomasz Łuczak, and Andrzej Rucinski, Random graphs, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley-Interscience, New York, 2000. MR 1782847, DOI 10.1002/9781118032718
Anders Johansson, Jeff Kahn, and Van Vu, Factors in random graphs, Random Structures Algorithms 33 (2008), no. 1, 1–28. MR 2428975, DOI 10.1002/rsa.20224
Jeff Kahn and Gil Kalai, Thresholds and expectation thresholds, Combin. Probab. Comput. 16 (2007), no. 3, 495–502. MR 2312440, DOI 10.1017/S0963548307008474
Gil Kalai, Boolean functions: influence, threshold and noise, European Congress of Mathematics, Eur. Math. Soc., Zürich, 2018, pp. 85–110. MR 3887762
Jeff Kahn, Bhargav Narayanan, and Jinyoung Park, The threshold for the square of a Hamilton cycle, Proc. Amer. Math. Soc. 149 (2021), no. 8, 3201–3208. MR 4273128, DOI 10.1090/proc/15419
Richard Montgomery, Spanning trees in random graphs, Adv. Math. 356 (2019), 106793, 92. MR 3998769, DOI 10.1016/j.aim.2019.106793
J. Park and H. T. Pham, On a conjecture of Talagrand on selector processes and a consequence on positive empirical processes, Preprint, arXiv:2204.10309 [math.PR].
Alexander A. Razborov, Bounded arithmetic and lower bounds in Boolean complexity, Feasible mathematics, II (Ithaca, NY, 1992) Progr. Comput. Sci. Appl. Logic, vol. 13, Birkhäuser Boston, Boston, MA, 1995, pp. 344–386. MR 1322282
Michel Talagrand, Selector processes on classes of sets, Probab. Theory Related Fields 135 (2006), no. 4, 471–486. MR 2240697, DOI 10.1007/s00440-004-0350-2
Michel Talagrand, Are many small sets explicitly small?, STOC’10—Proceedings of the 2010 ACM International Symposium on Theory of Computing, ACM, New York, 2010, pp. 13–35. MR 2743011
Michel Talagrand, Upper and lower bounds for stochastic processes, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 60, Springer, Heidelberg, 2014. Modern methods and classical problems. MR 3184689, DOI 10.1007/978-3-642-54075-2
M. Talagrand, personal communication, 2022.