求助PDF
{"title":"随机 3-Uniform 超图中松散汉密尔顿循环的转移","authors":"Kalina Petrova, Miloš Trujić","doi":"10.1002/rsa.21216","DOIUrl":null,"url":null,"abstract":"A loose Hamilton cycle in a hypergraph is a cyclic sequence of edges covering all vertices in which only every two consecutive edges intersect and do so in exactly one vertex. With Dirac's theorem in mind, it is natural to ask what minimum <span data-altimg=\"/cms/asset/96bf6800-4a53-4fcf-aa7b-a11ff802fdc1/rsa21216-math-0001.png\"></span><mjx-container ctxtmenu_counter=\"1746\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/rsa21216-math-0001.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=\"d\" 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:rsa21216:rsa21216-math-0001\" display=\"inline\" location=\"graphic/rsa21216-math-0001.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=\"d\" data-semantic-type=\"identifier\">d</mi></mrow>$$ d $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>-degree condition guarantees the existence of a loose Hamilton cycle in a <span data-altimg=\"/cms/asset/e14262ab-ac20-4578-be3f-bbe30fc08b73/rsa21216-math-0002.png\"></span><mjx-container ctxtmenu_counter=\"1747\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/rsa21216-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=\"k\" 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:rsa21216:rsa21216-math-0002\" display=\"inline\" location=\"graphic/rsa21216-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=\"k\" data-semantic-type=\"identifier\">k</mi></mrow>$$ k $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>-uniform hypergraph. For <span data-altimg=\"/cms/asset/03c2dae8-8d2b-4c90-a6ec-2587395b51ad/rsa21216-math-0003.png\"></span><mjx-container ctxtmenu_counter=\"1748\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/rsa21216-math-0003.png\"><mjx-semantics><mjx-mrow data-semantic-children=\"0,2\" data-semantic-content=\"1\" data-semantic- data-semantic-role=\"equality\" data-semantic-speech=\"k 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=\"equality\" 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:rsa21216:rsa21216-math-0003\" display=\"inline\" location=\"graphic/rsa21216-math-0003.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=\"equality\" data-semantic-speech=\"k 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\">k</mi><mo data-semantic-=\"\" data-semantic-operator=\"relseq,=\" data-semantic-parent=\"3\" data-semantic-role=\"equality\" 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>$$ k=3 $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> and each <span data-altimg=\"/cms/asset/e6b6f63c-2ad0-44eb-aa8e-f1176ef1a67c/rsa21216-math-0004.png\"></span><mjx-container ctxtmenu_counter=\"1749\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/rsa21216-math-0004.png\"><mjx-semantics><mjx-mrow data-semantic-children=\"0,8\" data-semantic-content=\"1\" data-semantic- data-semantic-role=\"element\" data-semantic-speech=\"d element of StartSet 1 comma 2 EndSet\" data-semantic-type=\"infixop\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"9\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\"infixop,∈\" data-semantic-parent=\"9\" data-semantic-role=\"element\" data-semantic-type=\"operator\" rspace=\"5\" space=\"5\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\"7\" data-semantic-content=\"2,6\" data-semantic- data-semantic-parent=\"9\" data-semantic-role=\"set collection\" data-semantic-type=\"fenced\"><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"8\" data-semantic-role=\"open\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\"3,4,5\" data-semantic-content=\"4\" data-semantic- data-semantic-parent=\"8\" data-semantic-role=\"sequence\" data-semantic-type=\"punctuated\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn><mjx-mo data-semantic- data-semantic-operator=\"punctuated\" data-semantic-parent=\"7\" data-semantic-role=\"comma\" data-semantic-type=\"punctuation\" rspace=\"3\" style=\"margin-left: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn></mjx-mrow><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"8\" data-semantic-role=\"close\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml display=\"inline\" unselectable=\"on\"><math altimg=\"urn:x-wiley:rsa:media:rsa21216:rsa21216-math-0004\" display=\"inline\" location=\"graphic/rsa21216-math-0004.png\" overflow=\"scroll\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-children=\"0,8\" data-semantic-content=\"1\" data-semantic-role=\"element\" data-semantic-speech=\"d element of StartSet 1 comma 2 EndSet\" data-semantic-type=\"infixop\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"9\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">d</mi><mo data-semantic-=\"\" data-semantic-operator=\"infixop,∈\" data-semantic-parent=\"9\" data-semantic-role=\"element\" data-semantic-type=\"operator\">∈</mo><mrow data-semantic-=\"\" data-semantic-children=\"7\" data-semantic-content=\"2,6\" data-semantic-parent=\"9\" data-semantic-role=\"set collection\" data-semantic-type=\"fenced\"><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"8\" data-semantic-role=\"open\" data-semantic-type=\"fence\" stretchy=\"false\">{</mo><mrow data-semantic-=\"\" data-semantic-children=\"3,4,5\" data-semantic-content=\"4\" data-semantic-parent=\"8\" data-semantic-role=\"sequence\" data-semantic-type=\"punctuated\"><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"7\" data-semantic-role=\"integer\" data-semantic-type=\"number\">1</mn><mo data-semantic-=\"\" data-semantic-operator=\"punctuated\" data-semantic-parent=\"7\" data-semantic-role=\"comma\" data-semantic-type=\"punctuation\">,</mo><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"7\" data-semantic-role=\"integer\" data-semantic-type=\"number\">2</mn></mrow><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"8\" data-semantic-role=\"close\" data-semantic-type=\"fence\" stretchy=\"false\">}</mo></mrow></mrow>$$ d\\in \\left\\{1,2\\right\\} $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>, the necessary and sufficient such condition is known precisely. We show that these results adhere to a ‘transference principle’ to their sparse random analogues. The proof combines several ideas from the graph setting and relies on the absorbing method. In particular, we employ a novel approach of Kwan and Ferber for finding absorbers in subgraphs of sparse hypergraphs via a contraction procedure. In the case of <span data-altimg=\"/cms/asset/2b7653d8-9656-49c2-ad12-7afeb4f3ff57/rsa21216-math-0005.png\"></span><mjx-container ctxtmenu_counter=\"1750\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/rsa21216-math-0005.png\"><mjx-semantics><mjx-mrow data-semantic-children=\"0,2\" data-semantic-content=\"1\" data-semantic- data-semantic-role=\"equality\" data-semantic-speech=\"d equals 2\" 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=\"equality\" 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:rsa21216:rsa21216-math-0005\" display=\"inline\" location=\"graphic/rsa21216-math-0005.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=\"equality\" data-semantic-speech=\"d equals 2\" 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\">d</mi><mo data-semantic-=\"\" data-semantic-operator=\"relseq,=\" data-semantic-parent=\"3\" data-semantic-role=\"equality\" 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\">2</mn></mrow>$$ d=2 $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>, our findings are asymptotically optimal.","PeriodicalId":20948,"journal":{"name":"Random Structures and Algorithms","volume":"34 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Transference for loose Hamilton cycles in random 3-uniform hypergraphs\",\"authors\":\"Kalina Petrova, Miloš Trujić\",\"doi\":\"10.1002/rsa.21216\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"A loose Hamilton cycle in a hypergraph is a cyclic sequence of edges covering all vertices in which only every two consecutive edges intersect and do so in exactly one vertex. With Dirac's theorem in mind, it is natural to ask what minimum <span data-altimg=\\\"/cms/asset/96bf6800-4a53-4fcf-aa7b-a11ff802fdc1/rsa21216-math-0001.png\\\"></span><mjx-container ctxtmenu_counter=\\\"1746\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\" location=\\\"graphic/rsa21216-math-0001.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=\\\"d\\\" 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:rsa21216:rsa21216-math-0001\\\" display=\\\"inline\\\" location=\\\"graphic/rsa21216-math-0001.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=\\\"d\\\" data-semantic-type=\\\"identifier\\\">d</mi></mrow>$$ d $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>-degree condition guarantees the existence of a loose Hamilton cycle in a <span data-altimg=\\\"/cms/asset/e14262ab-ac20-4578-be3f-bbe30fc08b73/rsa21216-math-0002.png\\\"></span><mjx-container ctxtmenu_counter=\\\"1747\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\" location=\\\"graphic/rsa21216-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=\\\"k\\\" 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:rsa21216:rsa21216-math-0002\\\" display=\\\"inline\\\" location=\\\"graphic/rsa21216-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=\\\"k\\\" data-semantic-type=\\\"identifier\\\">k</mi></mrow>$$ k $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>-uniform hypergraph. For <span data-altimg=\\\"/cms/asset/03c2dae8-8d2b-4c90-a6ec-2587395b51ad/rsa21216-math-0003.png\\\"></span><mjx-container ctxtmenu_counter=\\\"1748\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\" location=\\\"graphic/rsa21216-math-0003.png\\\"><mjx-semantics><mjx-mrow data-semantic-children=\\\"0,2\\\" data-semantic-content=\\\"1\\\" data-semantic- data-semantic-role=\\\"equality\\\" data-semantic-speech=\\\"k 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=\\\"equality\\\" 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:rsa21216:rsa21216-math-0003\\\" display=\\\"inline\\\" location=\\\"graphic/rsa21216-math-0003.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=\\\"equality\\\" data-semantic-speech=\\\"k 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\\\">k</mi><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"relseq,=\\\" data-semantic-parent=\\\"3\\\" data-semantic-role=\\\"equality\\\" 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>$$ k=3 $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> and each <span data-altimg=\\\"/cms/asset/e6b6f63c-2ad0-44eb-aa8e-f1176ef1a67c/rsa21216-math-0004.png\\\"></span><mjx-container ctxtmenu_counter=\\\"1749\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\" location=\\\"graphic/rsa21216-math-0004.png\\\"><mjx-semantics><mjx-mrow data-semantic-children=\\\"0,8\\\" data-semantic-content=\\\"1\\\" data-semantic- data-semantic-role=\\\"element\\\" data-semantic-speech=\\\"d element of StartSet 1 comma 2 EndSet\\\" data-semantic-type=\\\"infixop\\\"><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"9\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\\\"infixop,∈\\\" data-semantic-parent=\\\"9\\\" data-semantic-role=\\\"element\\\" data-semantic-type=\\\"operator\\\" rspace=\\\"5\\\" space=\\\"5\\\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\\\"7\\\" data-semantic-content=\\\"2,6\\\" data-semantic- data-semantic-parent=\\\"9\\\" data-semantic-role=\\\"set collection\\\" data-semantic-type=\\\"fenced\\\"><mjx-mo data-semantic- data-semantic-operator=\\\"fenced\\\" data-semantic-parent=\\\"8\\\" data-semantic-role=\\\"open\\\" data-semantic-type=\\\"fence\\\" style=\\\"margin-left: 0.056em; margin-right: 0.056em;\\\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\\\"3,4,5\\\" data-semantic-content=\\\"4\\\" data-semantic- data-semantic-parent=\\\"8\\\" data-semantic-role=\\\"sequence\\\" data-semantic-type=\\\"punctuated\\\"><mjx-mn data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic- data-semantic-parent=\\\"7\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\"><mjx-c></mjx-c></mjx-mn><mjx-mo data-semantic- data-semantic-operator=\\\"punctuated\\\" data-semantic-parent=\\\"7\\\" data-semantic-role=\\\"comma\\\" data-semantic-type=\\\"punctuation\\\" rspace=\\\"3\\\" style=\\\"margin-left: 0.056em;\\\"><mjx-c></mjx-c></mjx-mo><mjx-mn data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic- data-semantic-parent=\\\"7\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\"><mjx-c></mjx-c></mjx-mn></mjx-mrow><mjx-mo data-semantic- data-semantic-operator=\\\"fenced\\\" data-semantic-parent=\\\"8\\\" data-semantic-role=\\\"close\\\" data-semantic-type=\\\"fence\\\" style=\\\"margin-left: 0.056em; margin-right: 0.056em;\\\"><mjx-c></mjx-c></mjx-mo></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"urn:x-wiley:rsa:media:rsa21216:rsa21216-math-0004\\\" display=\\\"inline\\\" location=\\\"graphic/rsa21216-math-0004.png\\\" overflow=\\\"scroll\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><mrow data-semantic-=\\\"\\\" data-semantic-children=\\\"0,8\\\" data-semantic-content=\\\"1\\\" data-semantic-role=\\\"element\\\" data-semantic-speech=\\\"d element of StartSet 1 comma 2 EndSet\\\" data-semantic-type=\\\"infixop\\\"><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"9\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">d</mi><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"infixop,∈\\\" data-semantic-parent=\\\"9\\\" data-semantic-role=\\\"element\\\" data-semantic-type=\\\"operator\\\">∈</mo><mrow data-semantic-=\\\"\\\" data-semantic-children=\\\"7\\\" data-semantic-content=\\\"2,6\\\" data-semantic-parent=\\\"9\\\" data-semantic-role=\\\"set collection\\\" data-semantic-type=\\\"fenced\\\"><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"fenced\\\" data-semantic-parent=\\\"8\\\" data-semantic-role=\\\"open\\\" data-semantic-type=\\\"fence\\\" stretchy=\\\"false\\\">{</mo><mrow data-semantic-=\\\"\\\" data-semantic-children=\\\"3,4,5\\\" data-semantic-content=\\\"4\\\" data-semantic-parent=\\\"8\\\" data-semantic-role=\\\"sequence\\\" data-semantic-type=\\\"punctuated\\\"><mn data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic-parent=\\\"7\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\">1</mn><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"punctuated\\\" data-semantic-parent=\\\"7\\\" data-semantic-role=\\\"comma\\\" data-semantic-type=\\\"punctuation\\\">,</mo><mn data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic-parent=\\\"7\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\">2</mn></mrow><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"fenced\\\" data-semantic-parent=\\\"8\\\" data-semantic-role=\\\"close\\\" data-semantic-type=\\\"fence\\\" stretchy=\\\"false\\\">}</mo></mrow></mrow>$$ d\\\\in \\\\left\\\\{1,2\\\\right\\\\} $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>, the necessary and sufficient such condition is known precisely. We show that these results adhere to a ‘transference principle’ to their sparse random analogues. The proof combines several ideas from the graph setting and relies on the absorbing method. In particular, we employ a novel approach of Kwan and Ferber for finding absorbers in subgraphs of sparse hypergraphs via a contraction procedure. In the case of <span data-altimg=\\\"/cms/asset/2b7653d8-9656-49c2-ad12-7afeb4f3ff57/rsa21216-math-0005.png\\\"></span><mjx-container ctxtmenu_counter=\\\"1750\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\" location=\\\"graphic/rsa21216-math-0005.png\\\"><mjx-semantics><mjx-mrow data-semantic-children=\\\"0,2\\\" data-semantic-content=\\\"1\\\" data-semantic- data-semantic-role=\\\"equality\\\" data-semantic-speech=\\\"d equals 2\\\" 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=\\\"equality\\\" 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:rsa21216:rsa21216-math-0005\\\" display=\\\"inline\\\" location=\\\"graphic/rsa21216-math-0005.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=\\\"equality\\\" data-semantic-speech=\\\"d equals 2\\\" 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\\\">d</mi><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"relseq,=\\\" data-semantic-parent=\\\"3\\\" data-semantic-role=\\\"equality\\\" 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\\\">2</mn></mrow>$$ d=2 $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>, our findings are asymptotically optimal.\",\"PeriodicalId\":20948,\"journal\":{\"name\":\"Random Structures and Algorithms\",\"volume\":\"34 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-04-12\",\"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.21216\",\"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.21216","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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