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Thresholds versus fractional expectation-thresholds. (English) Zbl 1472.05132

Summary: Proving a conjecture of M. Talagrand [in: Proceedings of the 42nd annual ACM symposium on theory of computing, STOC ’10. Cambridge, MA, USA, June 5–8, 2010. New York, NY: Association for Computing Machinery (ACM). 13–36 (2010; Zbl 1293.60014)], a fractional version of the “expectation-threshold” conjecture of J. Kahn and G. Kalai [Comb. Probab. Comput. 16, No. 3, 495–502 (2007; Zbl 1118.05093)], we show that \(p_c(\mathcal{F})=O(q_f(\mathcal{F})\log\ell(\mathcal{F}))\) for any increasing family \(\mathcal{F}\) on a finite set \(X\), where \(p_c(\mathcal{F})\) and \(q_f(\mathcal{F})\) are the threshold and “fractional expectation-threshold” of \(\mathcal{F}\), and \(\ell(\mathcal{F})\) is the maximum size of a minimal member of \(\mathcal{F}\). This easily implies several heretofore difficult results and conjectures in probabilistic combinatorics, including thresholds for perfect hypergraph matchings [A. Johansson et al., Random Struct. Algorithms 33, No. 1, 1–28 (2008; Zbl 1146.05040)], bounded degree spanning trees [R. Montgomery, Adv. Math. 356, Article ID 106793, 92 p. (2019; Zbl 1421.05080)], and bounded degree graphs (new). We also resolve (and vastly extend) the “axial” version of the random multi-dimensional assignment problem (earlier considered by Martin-Mézard-Rivoire and A. Frieze and G. B. Sorkin [Random Struct. Algorithms 46, No. 1, 160–196 (2015; Zbl 1347.60141)]). Our approach builds on a recent breakthrough of R. Alweiss et al. [in: Proceedings of the 52nd annual ACM SIGACT symposium on theory of computing, STOC ’20, Chicago, IL, USA, June 22–26, 2020. New York, NY: Association for Computing Machinery (ACM). 624–630 (2020; Zbl 07298275)] on the Erdő-Rado “Sunflower Conjecture”.

MSC:

05C80 Random graphs (graph-theoretic aspects)
60D05 Geometric probability and stochastic geometry
60G15 Gaussian processes
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