关于高维分区的数量

IF 1.5 1区数学 Q1 MATHEMATICS

Proceedings of the London Mathematical Society Pub Date : 2024-02-14 DOI:10.1112/plms.12586

Cosmin Pohoata, Dmitrii Zakharov

{"title":"关于高维分区的数量","authors":"Cosmin Pohoata, Dmitrii Zakharov","doi":"10.1112/plms.12586","DOIUrl":null,"url":null,"abstract":"Let <mjx-container aria-label=\"upper P Subscript d Baseline left parenthesis n right parenthesis\" ctxtmenu_counter=\"0\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"2,6\" data-semantic-content=\"7,0\" data-semantic- data-semantic-role=\"simple function\" data-semantic-speech=\"upper P Subscript d Baseline left parenthesis n right parenthesis\" data-semantic-type=\"appl\"><mjx-msub data-semantic-children=\"0,1\" data-semantic- data-semantic-parent=\"8\" data-semantic-role=\"simple function\" data-semantic-type=\"subscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"2\" data-semantic-role=\"simple function\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em; margin-left: -0.109em;\"><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\" size=\"s\"><mjx-c></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"8\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\"4\" data-semantic-content=\"3,5\" data-semantic- data-semantic-parent=\"8\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" data-semantic-role=\"open\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" 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 aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/127a3c0f-7553-4810-ba2f-0ecc75bac01f/plms12586-math-0001.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"2,6\" data-semantic-content=\"7,0\" data-semantic-role=\"simple function\" data-semantic-speech=\"upper P Subscript d Baseline left parenthesis n right parenthesis\" data-semantic-type=\"appl\"><msub data-semantic-=\"\" data-semantic-children=\"0,1\" data-semantic-parent=\"8\" data-semantic-role=\"simple function\" data-semantic-type=\"subscript\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-operator=\"appl\" data-semantic-parent=\"2\" data-semantic-role=\"simple function\" data-semantic-type=\"identifier\">P</mi><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">d</mi></msub><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"appl\" data-semantic-parent=\"8\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\">⁡</mo><mrow data-semantic-=\"\" data-semantic-children=\"4\" data-semantic-content=\"3,5\" data-semantic-parent=\"8\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" data-semantic-role=\"open\" data-semantic-type=\"fence\" stretchy=\"false\">(</mo><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"6\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">n</mi><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" data-semantic-role=\"close\" data-semantic-type=\"fence\" stretchy=\"false\">)</mo></mrow></mrow>$P_{d}(n)$</annotation></semantics></math></mjx-assistive-mml></mjx-container> denote the number of <mjx-container aria-label=\"n times ellipsis times n d\" ctxtmenu_counter=\"1\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow data-semantic-children=\"7,2,10\" data-semantic-collapsed=\"(13 (c 11 12) 7 2 10)\" data-semantic- data-semantic-role=\"text\" data-semantic-speech=\"n times ellipsis times n d\" data-semantic-type=\"punctuated\"><mjx-mrow data-semantic-children=\"0\" data-semantic-content=\"1\" data-semantic- data-semantic-parent=\"13\" data-semantic-role=\"unknown\" data-semantic-type=\"postfixop\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\"postfixop,×\" data-semantic-parent=\"7\" data-semantic-role=\"unknown\" data-semantic-type=\"operator\" rspace=\"4\" space=\"4\"><mjx-c></mjx-c></mjx-mo></mjx-mrow><mjx-mtext data-semantic-annotation=\"clearspeak:unit\" data-semantic- data-semantic-parent=\"13\" data-semantic-role=\"ellipsis\" data-semantic-type=\"text\"><mjx-c></mjx-c></mjx-mtext><mjx-mrow data-semantic-children=\"9\" data-semantic-content=\"3\" data-semantic- data-semantic-parent=\"13\" data-semantic-role=\"unknown\" data-semantic-type=\"prefixop\"><mjx-mo data-semantic- data-semantic-operator=\"prefixop,×\" data-semantic-parent=\"10\" data-semantic-role=\"unknown\" data-semantic-type=\"operator\" rspace=\"4\" space=\"4\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-annotation=\"clearspeak:simple;clearspeak:unit\" data-semantic-children=\"4,6\" data-semantic-content=\"8\" data-semantic- data-semantic-parent=\"10\" data-semantic-role=\"implicit\" 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-mspace data-semantic- data-semantic-operator=\"infixop,⁢\" data-semantic-parent=\"9\" data-semantic-role=\"space\" data-semantic-type=\"operator\" style=\"width: 0.33em;\"></mjx-mspace><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-mrow></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/b52e2510-2dc9-4317-bff8-aa0abed97792/plms12586-math-0002.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-children=\"7,2,10\" data-semantic-collapsed=\"(13 (c 11 12) 7 2 10)\" data-semantic-role=\"text\" data-semantic-speech=\"n times ellipsis times n d\" data-semantic-type=\"punctuated\"><mrow data-semantic-=\"\" data-semantic-children=\"0\" data-semantic-content=\"1\" data-semantic-parent=\"13\" data-semantic-role=\"unknown\" data-semantic-type=\"postfixop\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"7\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">n</mi><mo data-semantic-=\"\" data-semantic-operator=\"postfixop,×\" data-semantic-parent=\"7\" data-semantic-role=\"unknown\" data-semantic-type=\"operator\">×</mo></mrow><mtext data-semantic-=\"\" data-semantic-annotation=\"clearspeak:unit\" data-semantic-parent=\"13\" data-semantic-role=\"ellipsis\" data-semantic-type=\"text\">…</mtext><mrow data-semantic-=\"\" data-semantic-children=\"9\" data-semantic-content=\"3\" data-semantic-parent=\"13\" data-semantic-role=\"unknown\" data-semantic-type=\"prefixop\"><mo data-semantic-=\"\" data-semantic-operator=\"prefixop,×\" data-semantic-parent=\"10\" data-semantic-role=\"unknown\" data-semantic-type=\"operator\">×</mo><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple;clearspeak:unit\" data-semantic-children=\"4,6\" data-semantic-content=\"8\" data-semantic-parent=\"10\" data-semantic-role=\"implicit\" 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\">n</mi><mspace data-semantic-=\"\" data-semantic-operator=\"infixop,⁢\" data-semantic-parent=\"9\" data-semantic-role=\"space\" data-semantic-type=\"operator\" width=\"0.33em\"></mspace><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></mrow></mrow></mrow>$n \\times \\ldots \\times n \\ d$</annotation></semantics></math></mjx-assistive-mml></mjx-container>-dimensional partitions with entries from <mjx-container aria-label=\"StartSet 0 comma 1 comma ellipsis comma n EndSet\" ctxtmenu_counter=\"2\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow data-semantic-children=\"9\" data-semantic-content=\"0,8\" data-semantic- data-semantic-role=\"set collection\" data-semantic-speech=\"StartSet 0 comma 1 comma ellipsis comma n EndSet\" data-semantic-type=\"fenced\"><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"10\" 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=\"1,2,3,4,5,6,7\" data-semantic-content=\"2,4,6\" data-semantic- data-semantic-parent=\"10\" data-semantic-role=\"sequence\" data-semantic-type=\"punctuated\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"9\" 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=\"9\" 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=\"9\" 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=\"9\" data-semantic-role=\"comma\" data-semantic-type=\"punctuation\" rspace=\"3\" style=\"margin-left: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mtext data-semantic-annotation=\"clearspeak:unit\" data-semantic- data-semantic-parent=\"9\" data-semantic-role=\"ellipsis\" data-semantic-type=\"text\"><mjx-c></mjx-c></mjx-mtext><mjx-mo data-semantic- data-semantic-operator=\"punctuated\" data-semantic-parent=\"9\" data-semantic-role=\"comma\" data-semantic-type=\"punctuation\" rspace=\"3\" style=\"margin-left: 0.056em;\"><mjx-c></mjx-c></mjx-mo><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-mrow><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"10\" 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-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/2fec70bd-ab1b-45d5-9465-eee2f2d10f09/plms12586-math-0003.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-children=\"9\" data-semantic-content=\"0,8\" data-semantic-role=\"set collection\" data-semantic-speech=\"StartSet 0 comma 1 comma ellipsis comma n EndSet\" data-semantic-type=\"fenced\"><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"10\" data-semantic-role=\"open\" data-semantic-type=\"fence\" stretchy=\"false\">{</mo><mrow data-semantic-=\"\" data-semantic-children=\"1,2,3,4,5,6,7\" data-semantic-content=\"2,4,6\" data-semantic-parent=\"10\" data-semantic-role=\"sequence\" data-semantic-type=\"punctuated\"><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"9\" data-semantic-role=\"integer\" data-semantic-type=\"number\">0</mn><mo data-semantic-=\"\" data-semantic-operator=\"punctuated\" data-semantic-parent=\"9\" data-semantic-role=\"comma\" data-semantic-type=\"punctuation\">,</mo><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"9\" data-semantic-role=\"integer\" data-semantic-type=\"number\">1</mn><mo data-semantic-=\"\" data-semantic-operator=\"punctuated\" data-semantic-parent=\"9\" data-semantic-role=\"comma\" data-semantic-type=\"punctuation\">,</mo><mtext data-semantic-=\"\" data-semantic-annotation=\"clearspeak:unit\" data-semantic-parent=\"9\" data-semantic-role=\"ellipsis\" data-semantic-type=\"text\">…</mtext><mo data-semantic-=\"\" data-semantic-operator=\"punctuated\" data-semantic-parent=\"9\" data-semantic-role=\"comma\" data-semantic-type=\"punctuation\">,</mo><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"9\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">n</mi></mrow><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"10\" data-semantic-role=\"close\" data-semantic-type=\"fence\" stretchy=\"false\">}</mo></mrow>$\\lbrace 0,1,\\ldots,n\\rbrace$</annotation></semantics></math></mjx-assistive-mml></mjx-container>. Building upon the works of Balogh–Treglown–Wagner and Noel–Scott–Sudakov, we show that when <mjx-container aria-label=\"d right arrow infinity\" ctxtmenu_counter=\"3\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow data-semantic-children=\"0,2\" data-semantic-content=\"1\" data-semantic- data-semantic-role=\"arrow\" data-semantic-speech=\"d right arrow infinity\" 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=\"arrow\" data-semantic-type=\"relation\" rspace=\"5\" space=\"5\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic- data-semantic-parent=\"3\" data-semantic-role=\"unknown\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/3b1d4e4a-749a-45c3-8d49-93976114ad3b/plms12586-math-0004.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-children=\"0,2\" data-semantic-content=\"1\" data-semantic-role=\"arrow\" data-semantic-speech=\"d right arrow infinity\" 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=\"arrow\" data-semantic-type=\"relation\">→</mo><mi data-semantic-=\"\" data-semantic-parent=\"3\" data-semantic-role=\"unknown\" data-semantic-type=\"identifier\">∞</mi></mrow>$d \\rightarrow \\infty$</annotation></semantics></math></mjx-assistive-mml></mjx-container>,","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"254 1","pages":""},"PeriodicalIF":1.5000,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the number of high-dimensional partitions\",\"authors\":\"Cosmin Pohoata, Dmitrii Zakharov\",\"doi\":\"10.1112/plms.12586\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let <mjx-container aria-label=\\\"upper P Subscript d Baseline left parenthesis n right parenthesis\\\" ctxtmenu_counter=\\\"0\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\"><mjx-semantics><mjx-mrow data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-children=\\\"2,6\\\" data-semantic-content=\\\"7,0\\\" data-semantic- data-semantic-role=\\\"simple function\\\" data-semantic-speech=\\\"upper P Subscript d Baseline left parenthesis n right parenthesis\\\" data-semantic-type=\\\"appl\\\"><mjx-msub data-semantic-children=\\\"0,1\\\" data-semantic- data-semantic-parent=\\\"8\\\" data-semantic-role=\\\"simple function\\\" data-semantic-type=\\\"subscript\\\"><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-operator=\\\"appl\\\" data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"simple function\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi><mjx-script style=\\\"vertical-align: -0.15em; margin-left: -0.109em;\\\"><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\\\" size=\\\"s\\\"><mjx-c></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo data-semantic-added=\\\"true\\\" data-semantic- data-semantic-operator=\\\"appl\\\" data-semantic-parent=\\\"8\\\" data-semantic-role=\\\"application\\\" data-semantic-type=\\\"punctuation\\\" style=\\\"margin-left: 0.056em; margin-right: 0.056em;\\\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\\\"4\\\" data-semantic-content=\\\"3,5\\\" data-semantic- data-semantic-parent=\\\"8\\\" data-semantic-role=\\\"leftright\\\" data-semantic-type=\\\"fenced\\\"><mjx-mo data-semantic- data-semantic-operator=\\\"fenced\\\" data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"open\\\" data-semantic-type=\\\"fence\\\" style=\\\"margin-left: 0.056em; margin-right: 0.056em;\\\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\\\"fenced\\\" data-semantic-parent=\\\"6\\\" 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 aria-hidden=\\\"true\\\" display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"/cms/asset/127a3c0f-7553-4810-ba2f-0ecc75bac01f/plms12586-math-0001.png\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><mrow data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-children=\\\"2,6\\\" data-semantic-content=\\\"7,0\\\" data-semantic-role=\\\"simple function\\\" data-semantic-speech=\\\"upper P Subscript d Baseline left parenthesis n right parenthesis\\\" data-semantic-type=\\\"appl\\\"><msub data-semantic-=\\\"\\\" data-semantic-children=\\\"0,1\\\" data-semantic-parent=\\\"8\\\" data-semantic-role=\\\"simple function\\\" data-semantic-type=\\\"subscript\\\"><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-operator=\\\"appl\\\" data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"simple function\\\" data-semantic-type=\\\"identifier\\\">P</mi><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">d</mi></msub><mo data-semantic-=\\\"\\\" data-semantic-added=\\\"true\\\" data-semantic-operator=\\\"appl\\\" data-semantic-parent=\\\"8\\\" data-semantic-role=\\\"application\\\" data-semantic-type=\\\"punctuation\\\">⁡</mo><mrow data-semantic-=\\\"\\\" data-semantic-children=\\\"4\\\" data-semantic-content=\\\"3,5\\\" data-semantic-parent=\\\"8\\\" data-semantic-role=\\\"leftright\\\" data-semantic-type=\\\"fenced\\\"><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"fenced\\\" data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"open\\\" data-semantic-type=\\\"fence\\\" stretchy=\\\"false\\\">(</mo><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">n</mi><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"fenced\\\" data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"close\\\" data-semantic-type=\\\"fence\\\" stretchy=\\\"false\\\">)</mo></mrow></mrow>$P_{d}(n)$</annotation></semantics></math></mjx-assistive-mml></mjx-container> denote the number of <mjx-container aria-label=\\\"n times ellipsis times n d\\\" ctxtmenu_counter=\\\"1\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\"><mjx-semantics><mjx-mrow data-semantic-children=\\\"7,2,10\\\" data-semantic-collapsed=\\\"(13 (c 11 12) 7 2 10)\\\" data-semantic- data-semantic-role=\\\"text\\\" data-semantic-speech=\\\"n times ellipsis times n d\\\" data-semantic-type=\\\"punctuated\\\"><mjx-mrow data-semantic-children=\\\"0\\\" data-semantic-content=\\\"1\\\" data-semantic- data-semantic-parent=\\\"13\\\" data-semantic-role=\\\"unknown\\\" data-semantic-type=\\\"postfixop\\\"><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"7\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\\\"postfixop,×\\\" data-semantic-parent=\\\"7\\\" data-semantic-role=\\\"unknown\\\" data-semantic-type=\\\"operator\\\" rspace=\\\"4\\\" space=\\\"4\\\"><mjx-c></mjx-c></mjx-mo></mjx-mrow><mjx-mtext data-semantic-annotation=\\\"clearspeak:unit\\\" data-semantic- data-semantic-parent=\\\"13\\\" data-semantic-role=\\\"ellipsis\\\" data-semantic-type=\\\"text\\\"><mjx-c></mjx-c></mjx-mtext><mjx-mrow data-semantic-children=\\\"9\\\" data-semantic-content=\\\"3\\\" data-semantic- data-semantic-parent=\\\"13\\\" data-semantic-role=\\\"unknown\\\" data-semantic-type=\\\"prefixop\\\"><mjx-mo data-semantic- data-semantic-operator=\\\"prefixop,×\\\" data-semantic-parent=\\\"10\\\" data-semantic-role=\\\"unknown\\\" data-semantic-type=\\\"operator\\\" rspace=\\\"4\\\" space=\\\"4\\\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-annotation=\\\"clearspeak:simple;clearspeak:unit\\\" data-semantic-children=\\\"4,6\\\" data-semantic-content=\\\"8\\\" data-semantic- data-semantic-parent=\\\"10\\\" data-semantic-role=\\\"implicit\\\" 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-mspace data-semantic- data-semantic-operator=\\\"infixop,⁢\\\" data-semantic-parent=\\\"9\\\" data-semantic-role=\\\"space\\\" data-semantic-type=\\\"operator\\\" style=\\\"width: 0.33em;\\\"></mjx-mspace><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-mrow></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\\\"true\\\" display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"/cms/asset/b52e2510-2dc9-4317-bff8-aa0abed97792/plms12586-math-0002.png\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><mrow data-semantic-=\\\"\\\" data-semantic-children=\\\"7,2,10\\\" data-semantic-collapsed=\\\"(13 (c 11 12) 7 2 10)\\\" data-semantic-role=\\\"text\\\" data-semantic-speech=\\\"n times ellipsis times n d\\\" data-semantic-type=\\\"punctuated\\\"><mrow data-semantic-=\\\"\\\" data-semantic-children=\\\"0\\\" data-semantic-content=\\\"1\\\" data-semantic-parent=\\\"13\\\" data-semantic-role=\\\"unknown\\\" data-semantic-type=\\\"postfixop\\\"><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"7\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">n</mi><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"postfixop,×\\\" data-semantic-parent=\\\"7\\\" data-semantic-role=\\\"unknown\\\" data-semantic-type=\\\"operator\\\">×</mo></mrow><mtext data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:unit\\\" data-semantic-parent=\\\"13\\\" data-semantic-role=\\\"ellipsis\\\" data-semantic-type=\\\"text\\\">…</mtext><mrow data-semantic-=\\\"\\\" data-semantic-children=\\\"9\\\" data-semantic-content=\\\"3\\\" data-semantic-parent=\\\"13\\\" data-semantic-role=\\\"unknown\\\" data-semantic-type=\\\"prefixop\\\"><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"prefixop,×\\\" data-semantic-parent=\\\"10\\\" data-semantic-role=\\\"unknown\\\" data-semantic-type=\\\"operator\\\">×</mo><mrow data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple;clearspeak:unit\\\" data-semantic-children=\\\"4,6\\\" data-semantic-content=\\\"8\\\" data-semantic-parent=\\\"10\\\" data-semantic-role=\\\"implicit\\\" 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\\\">n</mi><mspace data-semantic-=\\\"\\\" data-semantic-operator=\\\"infixop,⁢\\\" data-semantic-parent=\\\"9\\\" data-semantic-role=\\\"space\\\" data-semantic-type=\\\"operator\\\" width=\\\"0.33em\\\"></mspace><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></mrow></mrow></mrow>$n \\\\times \\\\ldots \\\\times n \\\\ d$</annotation></semantics></math></mjx-assistive-mml></mjx-container>-dimensional partitions with entries from <mjx-container aria-label=\\\"StartSet 0 comma 1 comma ellipsis comma n EndSet\\\" ctxtmenu_counter=\\\"2\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\"><mjx-semantics><mjx-mrow data-semantic-children=\\\"9\\\" data-semantic-content=\\\"0,8\\\" data-semantic- data-semantic-role=\\\"set collection\\\" data-semantic-speech=\\\"StartSet 0 comma 1 comma ellipsis comma n EndSet\\\" data-semantic-type=\\\"fenced\\\"><mjx-mo data-semantic- data-semantic-operator=\\\"fenced\\\" data-semantic-parent=\\\"10\\\" 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=\\\"1,2,3,4,5,6,7\\\" data-semantic-content=\\\"2,4,6\\\" data-semantic- data-semantic-parent=\\\"10\\\" data-semantic-role=\\\"sequence\\\" data-semantic-type=\\\"punctuated\\\"><mjx-mn data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic- data-semantic-parent=\\\"9\\\" 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=\\\"9\\\" 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=\\\"9\\\" 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=\\\"9\\\" data-semantic-role=\\\"comma\\\" data-semantic-type=\\\"punctuation\\\" rspace=\\\"3\\\" style=\\\"margin-left: 0.056em;\\\"><mjx-c></mjx-c></mjx-mo><mjx-mtext data-semantic-annotation=\\\"clearspeak:unit\\\" data-semantic- data-semantic-parent=\\\"9\\\" data-semantic-role=\\\"ellipsis\\\" data-semantic-type=\\\"text\\\"><mjx-c></mjx-c></mjx-mtext><mjx-mo data-semantic- data-semantic-operator=\\\"punctuated\\\" data-semantic-parent=\\\"9\\\" data-semantic-role=\\\"comma\\\" data-semantic-type=\\\"punctuation\\\" rspace=\\\"3\\\" style=\\\"margin-left: 0.056em;\\\"><mjx-c></mjx-c></mjx-mo><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-mrow><mjx-mo data-semantic- data-semantic-operator=\\\"fenced\\\" data-semantic-parent=\\\"10\\\" 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-semantics></mjx-math><mjx-assistive-mml aria-hidden=\\\"true\\\" display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"/cms/asset/2fec70bd-ab1b-45d5-9465-eee2f2d10f09/plms12586-math-0003.png\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><mrow data-semantic-=\\\"\\\" data-semantic-children=\\\"9\\\" data-semantic-content=\\\"0,8\\\" data-semantic-role=\\\"set collection\\\" data-semantic-speech=\\\"StartSet 0 comma 1 comma ellipsis comma n EndSet\\\" data-semantic-type=\\\"fenced\\\"><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"fenced\\\" data-semantic-parent=\\\"10\\\" data-semantic-role=\\\"open\\\" data-semantic-type=\\\"fence\\\" stretchy=\\\"false\\\">{</mo><mrow data-semantic-=\\\"\\\" data-semantic-children=\\\"1,2,3,4,5,6,7\\\" data-semantic-content=\\\"2,4,6\\\" data-semantic-parent=\\\"10\\\" data-semantic-role=\\\"sequence\\\" data-semantic-type=\\\"punctuated\\\"><mn data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic-parent=\\\"9\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\">0</mn><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"punctuated\\\" data-semantic-parent=\\\"9\\\" data-semantic-role=\\\"comma\\\" data-semantic-type=\\\"punctuation\\\">,</mo><mn data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic-parent=\\\"9\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\">1</mn><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"punctuated\\\" data-semantic-parent=\\\"9\\\" data-semantic-role=\\\"comma\\\" data-semantic-type=\\\"punctuation\\\">,</mo><mtext data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:unit\\\" data-semantic-parent=\\\"9\\\" data-semantic-role=\\\"ellipsis\\\" data-semantic-type=\\\"text\\\">…</mtext><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"punctuated\\\" data-semantic-parent=\\\"9\\\" data-semantic-role=\\\"comma\\\" data-semantic-type=\\\"punctuation\\\">,</mo><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"9\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">n</mi></mrow><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"fenced\\\" data-semantic-parent=\\\"10\\\" data-semantic-role=\\\"close\\\" data-semantic-type=\\\"fence\\\" stretchy=\\\"false\\\">}</mo></mrow>$\\\\lbrace 0,1,\\\\ldots,n\\\\rbrace$</annotation></semantics></math></mjx-assistive-mml></mjx-container>. Building upon the works of Balogh–Treglown–Wagner and Noel–Scott–Sudakov, we show that when <mjx-container aria-label=\\\"d right arrow infinity\\\" ctxtmenu_counter=\\\"3\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\"><mjx-semantics><mjx-mrow data-semantic-children=\\\"0,2\\\" data-semantic-content=\\\"1\\\" data-semantic- data-semantic-role=\\\"arrow\\\" data-semantic-speech=\\\"d right arrow infinity\\\" 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=\\\"arrow\\\" data-semantic-type=\\\"relation\\\" rspace=\\\"5\\\" space=\\\"5\\\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic- data-semantic-parent=\\\"3\\\" data-semantic-role=\\\"unknown\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\\\"true\\\" display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"/cms/asset/3b1d4e4a-749a-45c3-8d49-93976114ad3b/plms12586-math-0004.png\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><mrow data-semantic-=\\\"\\\" data-semantic-children=\\\"0,2\\\" data-semantic-content=\\\"1\\\" data-semantic-role=\\\"arrow\\\" data-semantic-speech=\\\"d right arrow infinity\\\" 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=\\\"arrow\\\" data-semantic-type=\\\"relation\\\">→</mo><mi data-semantic-=\\\"\\\" data-semantic-parent=\\\"3\\\" data-semantic-role=\\\"unknown\\\" data-semantic-type=\\\"identifier\\\">∞</mi></mrow>$d \\\\rightarrow \\\\infty$</annotation></semantics></math></mjx-assistive-mml></mjx-container>,\",\"PeriodicalId\":49667,\"journal\":{\"name\":\"Proceedings of the London Mathematical Society\",\"volume\":\"254 1\",\"pages\":\"\"},\"PeriodicalIF\":1.5000,\"publicationDate\":\"2024-02-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the London Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1112/plms.12586\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the London Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1112/plms.12586","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

摘要

让 Pd(n)$P_{d}(n)$ 表示条目来自 {0,1,...n}$\lbrace 0,1,\ldots,n\rbrace$ 的 n×...×nd$n \times \ldots \times n \ d$ 维分区的个数。在 Balogh-Treglown-Wagner 和 Noel-Scott-Sudakov 的研究基础上，我们证明了当 d→∞$d \rightarrow \infty$ 时、

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On the number of high-dimensional partitions

Let

P_{d} (n) $P_{d}(n)$

denote the number of

n \times … \times n d $n \times \ldots \times n \ d$

-dimensional partitions with entries from

{0, 1, …, n} $\lbrace 0,1,\ldots,n\rbrace$

. Building upon the works of Balogh–Treglown–Wagner and Noel–Scott–Sudakov, we show that when

d \to \infty $d \rightarrow \infty$

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来源期刊

Proceedings of the London Mathematical Society 数学-数学

CiteScore

2.90

自引率

0.00%

发文量

审稿时长

6-12 weeks

期刊介绍： The Proceedings of the London Mathematical Society is the flagship journal of the LMS. It publishes articles of the highest quality and significance across a broad range of mathematics. There are no page length restrictions for submitted papers. The Proceedings has its own Editorial Board separate from that of the Journal, Bulletin and Transactions of the LMS.