求助PDF
{"title":"小群因子的命中时间","authors":"Annika Heckel, Marc Kaufmann, Noela Müller, Matija Pasch","doi":"10.1002/rsa.21218","DOIUrl":null,"url":null,"abstract":"In [Trans. Am. Math. Soc. 375 (2022), no. 1, 627–668], Kahn gave the strongest possible, affirmative, answer to Shamir's problem, which had been open since the late 1970s: Let <span data-altimg=\"/cms/asset/7b4fb8d9-d0bb-4887-b2d9-9622142ef5ee/rsa21218-math-0001.png\"></span><mjx-container ctxtmenu_counter=\"1925\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/rsa21218-math-0001.png\"><mjx-semantics><mjx-mrow data-semantic-children=\"0,2\" data-semantic-content=\"1\" data-semantic- data-semantic-role=\"inequality\" data-semantic-speech=\"r greater than or slanted equals 3\" data-semantic-type=\"relseq\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\"relseq,⩾\" data-semantic-parent=\"3\" data-semantic-role=\"inequality\" data-semantic-type=\"relation\" rspace=\"5\" space=\"5\"><mjx-c></mjx-c></mjx-mo><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"3\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml display=\"inline\" unselectable=\"on\"><math altimg=\"urn:x-wiley:rsa:media:rsa21218:rsa21218-math-0001\" display=\"inline\" location=\"graphic/rsa21218-math-0001.png\" overflow=\"scroll\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-children=\"0,2\" data-semantic-content=\"1\" data-semantic-role=\"inequality\" data-semantic-speech=\"r greater than or slanted equals 3\" data-semantic-type=\"relseq\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">r</mi><mo data-semantic-=\"\" data-semantic-operator=\"relseq,⩾\" data-semantic-parent=\"3\" data-semantic-role=\"inequality\" data-semantic-type=\"relation\">⩾</mo><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"3\" data-semantic-role=\"integer\" data-semantic-type=\"number\">3</mn></mrow>$$ r\\geqslant 3 $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> and let <span data-altimg=\"/cms/asset/30e5ceff-d56d-4eb6-b13d-70f6602bea8e/rsa21218-math-0002.png\"></span><mjx-container ctxtmenu_counter=\"1926\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/rsa21218-math-0002.png\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"n\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml display=\"inline\" unselectable=\"on\"><math altimg=\"urn:x-wiley:rsa:media:rsa21218:rsa21218-math-0002\" display=\"inline\" location=\"graphic/rsa21218-math-0002.png\" overflow=\"scroll\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"latinletter\" data-semantic-speech=\"n\" data-semantic-type=\"identifier\">n</mi></mrow>$$ n $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> be divisible by <span data-altimg=\"/cms/asset/464e97c3-746f-4713-a2c0-ee34c315477f/rsa21218-math-0003.png\"></span><mjx-container ctxtmenu_counter=\"1927\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/rsa21218-math-0003.png\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"r\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml display=\"inline\" unselectable=\"on\"><math altimg=\"urn:x-wiley:rsa:media:rsa21218:rsa21218-math-0003\" display=\"inline\" location=\"graphic/rsa21218-math-0003.png\" overflow=\"scroll\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"latinletter\" data-semantic-speech=\"r\" data-semantic-type=\"identifier\">r</mi></mrow>$$ r $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>. Then, in the random <span data-altimg=\"/cms/asset/a251d35e-c54c-47e2-a16a-eb53fd9eea30/rsa21218-math-0004.png\"></span><mjx-container ctxtmenu_counter=\"1928\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/rsa21218-math-0004.png\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"r\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml display=\"inline\" unselectable=\"on\"><math altimg=\"urn:x-wiley:rsa:media:rsa21218:rsa21218-math-0004\" display=\"inline\" location=\"graphic/rsa21218-math-0004.png\" overflow=\"scroll\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"latinletter\" data-semantic-speech=\"r\" data-semantic-type=\"identifier\">r</mi></mrow>$$ r $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>-uniform hypergraph process on <span data-altimg=\"/cms/asset/af6deca9-7059-4b0b-879f-12e8f5b88639/rsa21218-math-0005.png\"></span><mjx-container ctxtmenu_counter=\"1929\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/rsa21218-math-0005.png\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"n\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml display=\"inline\" unselectable=\"on\"><math altimg=\"urn:x-wiley:rsa:media:rsa21218:rsa21218-math-0005\" display=\"inline\" location=\"graphic/rsa21218-math-0005.png\" overflow=\"scroll\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"latinletter\" data-semantic-speech=\"n\" data-semantic-type=\"identifier\">n</mi></mrow>$$ n $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> vertices, as soon as the last isolated vertex disappears, a perfect matching emerges. In the present work, we prove the analogue of this result for clique factors in the random graph process: at the time that the last vertex joins a copy of the complete graph <span data-altimg=\"/cms/asset/751a2dfc-ef84-4446-9471-4bdaa6837f11/rsa21218-math-0006.png\"></span><mjx-container ctxtmenu_counter=\"1930\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/rsa21218-math-0006.png\"><mjx-semantics><mjx-mrow><mjx-msub data-semantic-children=\"0,1\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper K Subscript r\" data-semantic-type=\"subscript\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-script style=\"vertical-align: -0.15em; margin-left: -0.04em;\"><mjx-mrow size=\"s\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-script></mjx-msub></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml display=\"inline\" unselectable=\"on\"><math altimg=\"urn:x-wiley:rsa:media:rsa21218:rsa21218-math-0006\" display=\"inline\" location=\"graphic/rsa21218-math-0006.png\" overflow=\"scroll\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub data-semantic-=\"\" data-semantic-children=\"0,1\" data-semantic-role=\"latinletter\" data-semantic-speech=\"upper K Subscript r\" data-semantic-type=\"subscript\"><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">K</mi></mrow><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">r</mi></mrow></msub></mrow>$$ {K}_r $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>, the random graph process contains a <span data-altimg=\"/cms/asset/47fde01d-5e90-4b22-91a5-7a12fd7a7695/rsa21218-math-0007.png\"></span><mjx-container ctxtmenu_counter=\"1931\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/rsa21218-math-0007.png\"><mjx-semantics><mjx-mrow><mjx-msub data-semantic-children=\"0,1\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper K Subscript r\" data-semantic-type=\"subscript\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-script style=\"vertical-align: -0.15em; margin-left: -0.04em;\"><mjx-mrow size=\"s\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-script></mjx-msub></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml display=\"inline\" unselectable=\"on\"><math altimg=\"urn:x-wiley:rsa:media:rsa21218:rsa21218-math-0007\" display=\"inline\" location=\"graphic/rsa21218-math-0007.png\" overflow=\"scroll\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub data-semantic-=\"\" data-semantic-children=\"0,1\" data-semantic-role=\"latinletter\" data-semantic-speech=\"upper K Subscript r\" data-semantic-type=\"subscript\"><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">K</mi></mrow><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">r</mi></mrow></msub></mrow>$$ {K}_r $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>-factor. Our proof draws on a novel sequence of couplings which embeds the random hypergraph process into the cliques of the random graph process. An analogous result is proved for clique factors in the <span data-altimg=\"/cms/asset/511928f4-aaad-4c95-8118-61bb5bb45b96/rsa21218-math-0008.png\"></span><mjx-container ctxtmenu_counter=\"1932\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/rsa21218-math-0008.png\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"s\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml display=\"inline\" unselectable=\"on\"><math altimg=\"urn:x-wiley:rsa:media:rsa21218:rsa21218-math-0008\" display=\"inline\" location=\"graphic/rsa21218-math-0008.png\" overflow=\"scroll\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"latinletter\" data-semantic-speech=\"s\" data-semantic-type=\"identifier\">s</mi></mrow>$$ s $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>-uniform hypergraph process (<span data-altimg=\"/cms/asset/8402910b-e94d-4f6f-88a6-03d9cdfe679e/rsa21218-math-0009.png\"></span><mjx-container ctxtmenu_counter=\"1933\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/rsa21218-math-0009.png\"><mjx-semantics><mjx-mrow data-semantic-children=\"0,2\" data-semantic-content=\"1\" data-semantic- data-semantic-role=\"inequality\" data-semantic-speech=\"s greater than or slanted equals 3\" data-semantic-type=\"relseq\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\"relseq,⩾\" data-semantic-parent=\"3\" data-semantic-role=\"inequality\" data-semantic-type=\"relation\" rspace=\"5\" space=\"5\"><mjx-c></mjx-c></mjx-mo><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"3\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml display=\"inline\" unselectable=\"on\"><math altimg=\"urn:x-wiley:rsa:media:rsa21218:rsa21218-math-0009\" display=\"inline\" location=\"graphic/rsa21218-math-0009.png\" overflow=\"scroll\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-children=\"0,2\" data-semantic-content=\"1\" data-semantic-role=\"inequality\" data-semantic-speech=\"s greater than or slanted equals 3\" data-semantic-type=\"relseq\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">s</mi><mo data-semantic-=\"\" data-semantic-operator=\"relseq,⩾\" data-semantic-parent=\"3\" data-semantic-role=\"inequality\" data-semantic-type=\"relation\">⩾</mo><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"3\" data-semantic-role=\"integer\" data-semantic-type=\"number\">3</mn></mrow>$$ s\\geqslant 3 $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>).","PeriodicalId":20948,"journal":{"name":"Random Structures and Algorithms","volume":"34 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The hitting time of clique factors\",\"authors\":\"Annika Heckel, Marc Kaufmann, Noela Müller, Matija Pasch\",\"doi\":\"10.1002/rsa.21218\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In [Trans. Am. Math. Soc. 375 (2022), no. 1, 627–668], Kahn gave the strongest possible, affirmative, answer to Shamir's problem, which had been open since the late 1970s: Let <span data-altimg=\\\"/cms/asset/7b4fb8d9-d0bb-4887-b2d9-9622142ef5ee/rsa21218-math-0001.png\\\"></span><mjx-container ctxtmenu_counter=\\\"1925\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\" location=\\\"graphic/rsa21218-math-0001.png\\\"><mjx-semantics><mjx-mrow data-semantic-children=\\\"0,2\\\" data-semantic-content=\\\"1\\\" data-semantic- data-semantic-role=\\\"inequality\\\" data-semantic-speech=\\\"r greater than or slanted equals 3\\\" data-semantic-type=\\\"relseq\\\"><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"3\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\\\"relseq,⩾\\\" data-semantic-parent=\\\"3\\\" data-semantic-role=\\\"inequality\\\" data-semantic-type=\\\"relation\\\" rspace=\\\"5\\\" space=\\\"5\\\"><mjx-c></mjx-c></mjx-mo><mjx-mn data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic- data-semantic-parent=\\\"3\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\"><mjx-c></mjx-c></mjx-mn></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"urn:x-wiley:rsa:media:rsa21218:rsa21218-math-0001\\\" display=\\\"inline\\\" location=\\\"graphic/rsa21218-math-0001.png\\\" overflow=\\\"scroll\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><mrow data-semantic-=\\\"\\\" data-semantic-children=\\\"0,2\\\" data-semantic-content=\\\"1\\\" data-semantic-role=\\\"inequality\\\" data-semantic-speech=\\\"r greater than or slanted equals 3\\\" data-semantic-type=\\\"relseq\\\"><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"3\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">r</mi><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"relseq,⩾\\\" data-semantic-parent=\\\"3\\\" data-semantic-role=\\\"inequality\\\" data-semantic-type=\\\"relation\\\">⩾</mo><mn data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic-parent=\\\"3\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\">3</mn></mrow>$$ r\\\\geqslant 3 $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> and let <span data-altimg=\\\"/cms/asset/30e5ceff-d56d-4eb6-b13d-70f6602bea8e/rsa21218-math-0002.png\\\"></span><mjx-container ctxtmenu_counter=\\\"1926\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\" location=\\\"graphic/rsa21218-math-0002.png\\\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"n\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"urn:x-wiley:rsa:media:rsa21218:rsa21218-math-0002\\\" display=\\\"inline\\\" location=\\\"graphic/rsa21218-math-0002.png\\\" overflow=\\\"scroll\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><mrow><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"n\\\" data-semantic-type=\\\"identifier\\\">n</mi></mrow>$$ n $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> be divisible by <span data-altimg=\\\"/cms/asset/464e97c3-746f-4713-a2c0-ee34c315477f/rsa21218-math-0003.png\\\"></span><mjx-container ctxtmenu_counter=\\\"1927\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\" location=\\\"graphic/rsa21218-math-0003.png\\\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"r\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"urn:x-wiley:rsa:media:rsa21218:rsa21218-math-0003\\\" display=\\\"inline\\\" location=\\\"graphic/rsa21218-math-0003.png\\\" overflow=\\\"scroll\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><mrow><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"r\\\" data-semantic-type=\\\"identifier\\\">r</mi></mrow>$$ r $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>. Then, in the random <span data-altimg=\\\"/cms/asset/a251d35e-c54c-47e2-a16a-eb53fd9eea30/rsa21218-math-0004.png\\\"></span><mjx-container ctxtmenu_counter=\\\"1928\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\" location=\\\"graphic/rsa21218-math-0004.png\\\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"r\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"urn:x-wiley:rsa:media:rsa21218:rsa21218-math-0004\\\" display=\\\"inline\\\" location=\\\"graphic/rsa21218-math-0004.png\\\" overflow=\\\"scroll\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><mrow><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"r\\\" data-semantic-type=\\\"identifier\\\">r</mi></mrow>$$ r $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>-uniform hypergraph process on <span data-altimg=\\\"/cms/asset/af6deca9-7059-4b0b-879f-12e8f5b88639/rsa21218-math-0005.png\\\"></span><mjx-container ctxtmenu_counter=\\\"1929\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\" location=\\\"graphic/rsa21218-math-0005.png\\\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"n\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"urn:x-wiley:rsa:media:rsa21218:rsa21218-math-0005\\\" display=\\\"inline\\\" location=\\\"graphic/rsa21218-math-0005.png\\\" overflow=\\\"scroll\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><mrow><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"n\\\" data-semantic-type=\\\"identifier\\\">n</mi></mrow>$$ n $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> vertices, as soon as the last isolated vertex disappears, a perfect matching emerges. In the present work, we prove the analogue of this result for clique factors in the random graph process: at the time that the last vertex joins a copy of the complete graph <span data-altimg=\\\"/cms/asset/751a2dfc-ef84-4446-9471-4bdaa6837f11/rsa21218-math-0006.png\\\"></span><mjx-container ctxtmenu_counter=\\\"1930\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\" location=\\\"graphic/rsa21218-math-0006.png\\\"><mjx-semantics><mjx-mrow><mjx-msub data-semantic-children=\\\"0,1\\\" data-semantic- data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"upper K Subscript r\\\" data-semantic-type=\\\"subscript\\\"><mjx-mrow><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-script style=\\\"vertical-align: -0.15em; margin-left: -0.04em;\\\"><mjx-mrow size=\\\"s\\\"><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-script></mjx-msub></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"urn:x-wiley:rsa:media:rsa21218:rsa21218-math-0006\\\" display=\\\"inline\\\" location=\\\"graphic/rsa21218-math-0006.png\\\" overflow=\\\"scroll\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><mrow><msub data-semantic-=\\\"\\\" data-semantic-children=\\\"0,1\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"upper K Subscript r\\\" data-semantic-type=\\\"subscript\\\"><mrow><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">K</mi></mrow><mrow><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">r</mi></mrow></msub></mrow>$$ {K}_r $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>, the random graph process contains a <span data-altimg=\\\"/cms/asset/47fde01d-5e90-4b22-91a5-7a12fd7a7695/rsa21218-math-0007.png\\\"></span><mjx-container ctxtmenu_counter=\\\"1931\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\" location=\\\"graphic/rsa21218-math-0007.png\\\"><mjx-semantics><mjx-mrow><mjx-msub data-semantic-children=\\\"0,1\\\" data-semantic- data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"upper K Subscript r\\\" data-semantic-type=\\\"subscript\\\"><mjx-mrow><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-script style=\\\"vertical-align: -0.15em; margin-left: -0.04em;\\\"><mjx-mrow size=\\\"s\\\"><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-script></mjx-msub></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"urn:x-wiley:rsa:media:rsa21218:rsa21218-math-0007\\\" display=\\\"inline\\\" location=\\\"graphic/rsa21218-math-0007.png\\\" overflow=\\\"scroll\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><mrow><msub data-semantic-=\\\"\\\" data-semantic-children=\\\"0,1\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"upper K Subscript r\\\" data-semantic-type=\\\"subscript\\\"><mrow><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">K</mi></mrow><mrow><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">r</mi></mrow></msub></mrow>$$ {K}_r $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>-factor. Our proof draws on a novel sequence of couplings which embeds the random hypergraph process into the cliques of the random graph process. An analogous result is proved for clique factors in the <span data-altimg=\\\"/cms/asset/511928f4-aaad-4c95-8118-61bb5bb45b96/rsa21218-math-0008.png\\\"></span><mjx-container ctxtmenu_counter=\\\"1932\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\" location=\\\"graphic/rsa21218-math-0008.png\\\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"s\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"urn:x-wiley:rsa:media:rsa21218:rsa21218-math-0008\\\" display=\\\"inline\\\" location=\\\"graphic/rsa21218-math-0008.png\\\" overflow=\\\"scroll\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><mrow><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"s\\\" data-semantic-type=\\\"identifier\\\">s</mi></mrow>$$ s $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>-uniform hypergraph process (<span data-altimg=\\\"/cms/asset/8402910b-e94d-4f6f-88a6-03d9cdfe679e/rsa21218-math-0009.png\\\"></span><mjx-container ctxtmenu_counter=\\\"1933\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\" location=\\\"graphic/rsa21218-math-0009.png\\\"><mjx-semantics><mjx-mrow data-semantic-children=\\\"0,2\\\" data-semantic-content=\\\"1\\\" data-semantic- data-semantic-role=\\\"inequality\\\" data-semantic-speech=\\\"s greater than or slanted equals 3\\\" data-semantic-type=\\\"relseq\\\"><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"3\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\\\"relseq,⩾\\\" data-semantic-parent=\\\"3\\\" data-semantic-role=\\\"inequality\\\" data-semantic-type=\\\"relation\\\" rspace=\\\"5\\\" space=\\\"5\\\"><mjx-c></mjx-c></mjx-mo><mjx-mn data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic- data-semantic-parent=\\\"3\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\"><mjx-c></mjx-c></mjx-mn></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"urn:x-wiley:rsa:media:rsa21218:rsa21218-math-0009\\\" display=\\\"inline\\\" location=\\\"graphic/rsa21218-math-0009.png\\\" overflow=\\\"scroll\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><mrow data-semantic-=\\\"\\\" data-semantic-children=\\\"0,2\\\" data-semantic-content=\\\"1\\\" data-semantic-role=\\\"inequality\\\" data-semantic-speech=\\\"s greater than or slanted equals 3\\\" data-semantic-type=\\\"relseq\\\"><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"3\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">s</mi><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"relseq,⩾\\\" data-semantic-parent=\\\"3\\\" data-semantic-role=\\\"inequality\\\" data-semantic-type=\\\"relation\\\">⩾</mo><mn data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic-parent=\\\"3\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\">3</mn></mrow>$$ s\\\\geqslant 3 $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>).\",\"PeriodicalId\":20948,\"journal\":{\"name\":\"Random Structures and Algorithms\",\"volume\":\"34 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-03-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Random Structures and Algorithms\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1002/rsa.21218\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Random Structures and Algorithms","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1002/rsa.21218","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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