Kahn–Kalai猜想的一个证明

IF 3.5 1区数学 Q1 MATHEMATICS

Journal of the American Mathematical Society Pub Date : 2023-08-07 DOI:10.1090/jams/1028

Jin-woo Park, Huye^n Pham

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Comput. 16 (2007), pp. 495–502], we show that for any increasing property <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper F\">\n <mml:semantics>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">F</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathcal {F}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> on a finite set <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\n <mml:semantics>\n <mml:mi>X</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">X</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, <disp-formula content-type=\"math/mathml\">\n\\[\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p Subscript c Baseline left-parenthesis script upper F right-parenthesis equals upper O left-parenthesis q left-parenthesis script upper F right-parenthesis log script l left-parenthesis script upper F right-parenthesis right-parenthesis comma\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>p</mml:mi>\n <mml:mi>c</mml:mi>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">F</mml:mi>\n </mml:mrow>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>=</mml:mo>\n <mml:mi>O</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>q</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">F</mml:mi>\n </mml:mrow>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mi>log</mml:mi>\n <mml:mo>⁡</mml:mo>\n <mml:mi>ℓ</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">F</mml:mi>\n </mml:mrow>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>,</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">p_c(\\mathcal {F})=O(q(\\mathcal {F})\\log \\ell (\\mathcal {F})),</mml:annotation>\n </mml:semantics>\n</mml:math>\n\\]\n</disp-formula> where <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p Subscript c Baseline left-parenthesis script upper F right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>p</mml:mi>\n <mml:mi>c</mml:mi>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">F</mml:mi>\n </mml:mrow>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">p_c(\\mathcal {F})</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"q left-parenthesis script upper F right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>q</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">F</mml:mi>\n </mml:mrow>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">q(\\mathcal {F})</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> are the threshold and “expectation threshold” of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper F\">\n <mml:semantics>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">F</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathcal {F}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script l left-parenthesis script upper F right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>ℓ</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">F</mml:mi>\n </mml:mrow>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\ell (\\mathcal {F})</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is the maximum of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2\">\n <mml:semantics>\n <mml:mn>2</mml:mn>\n <mml:annotation encoding=\"application/x-tex\">2</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and the maximum size of a minimal member of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper F\">\n <mml:semantics>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">F</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathcal {F}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>.</p>","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"1 1","pages":""},"PeriodicalIF":3.5000,"publicationDate":"2023-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"A proof of the Kahn–Kalai conjecture\",\"authors\":\"Jin-woo Park, Huye^n Pham\",\"doi\":\"10.1090/jams/1028\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Proving the “expectation-threshold” conjecture of Kahn and Kalai [Combin. 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Baseline left-parenthesis script upper F right-parenthesis equals upper O left-parenthesis q left-parenthesis script upper F right-parenthesis log script l left-parenthesis script upper F right-parenthesis right-parenthesis comma\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:msub>\\n <mml:mi>p</mml:mi>\\n <mml:mi>c</mml:mi>\\n </mml:msub>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi class=\\\"MJX-tex-caligraphic\\\" mathvariant=\\\"script\\\">F</mml:mi>\\n </mml:mrow>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:mo>=</mml:mo>\\n <mml:mi>O</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>q</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi class=\\\"MJX-tex-caligraphic\\\" mathvariant=\\\"script\\\">F</mml:mi>\\n </mml:mrow>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:mi>log</mml:mi>\\n <mml:mo>⁡</mml:mo>\\n <mml:mi>ℓ</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi class=\\\"MJX-tex-caligraphic\\\" mathvariant=\\\"script\\\">F</mml:mi>\\n </mml:mrow>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:mo>,</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">p_c(\\\\mathcal {F})=O(q(\\\\mathcal {F})\\\\log \\\\ell (\\\\mathcal {F})),</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n\\\\]\\n</disp-formula> where <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"p Subscript c Baseline left-parenthesis script upper F right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:msub>\\n <mml:mi>p</mml:mi>\\n <mml:mi>c</mml:mi>\\n </mml:msub>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi class=\\\"MJX-tex-caligraphic\\\" mathvariant=\\\"script\\\">F</mml:mi>\\n </mml:mrow>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">p_c(\\\\mathcal {F})</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> and <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"q left-parenthesis script upper F right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>q</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi class=\\\"MJX-tex-caligraphic\\\" mathvariant=\\\"script\\\">F</mml:mi>\\n </mml:mrow>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">q(\\\\mathcal {F})</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> are the threshold and “expectation threshold” of <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"script upper F\\\">\\n <mml:semantics>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi class=\\\"MJX-tex-caligraphic\\\" mathvariant=\\\"script\\\">F</mml:mi>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathcal {F}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>, and <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"script l left-parenthesis script upper F right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>ℓ</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi class=\\\"MJX-tex-caligraphic\\\" mathvariant=\\\"script\\\">F</mml:mi>\\n </mml:mrow>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\ell (\\\\mathcal {F})</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> is the maximum of <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"2\\\">\\n <mml:semantics>\\n <mml:mn>2</mml:mn>\\n <mml:annotation encoding=\\\"application/x-tex\\\">2</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> and the maximum size of a minimal member of <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"script upper F\\\">\\n <mml:semantics>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi class=\\\"MJX-tex-caligraphic\\\" mathvariant=\\\"script\\\">F</mml:mi>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathcal {F}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>.</p>\",\"PeriodicalId\":54764,\"journal\":{\"name\":\"Journal of the American Mathematical 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引用次数: 2

摘要

证明Kahn和Kalai的“期望阈值”猜想[Combin.Probab.Comput.16（2007），pp.495-502]，我们证明了对于有限集X上的任何增加性质F\mathcal｛F｝，\[pc（F）=O（q（F）log⁡ ℓ （F）），p_c（\mathcal｛F｝）=O（q（\mathcal{F｝，和ℓ （F）\ell（\mathcal｛F｝）是2的最大值，也是F\mathcal{F}的最小成员的最大大小。

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A proof of the Kahn–Kalai conjecture

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

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来源期刊

Journal of the American Mathematical Society 数学-数学

CiteScore

7.60

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： All articles submitted to this journal are peer-reviewed. The AMS has a single blind peer-review process in which the reviewers know who the authors of the manuscript are, but the authors do not have access to the information on who the peer reviewers are. This journal is devoted to research articles of the highest quality in all areas of pure and applied mathematics.