[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-272522-105":3,"detail-sidebar-cat-0-en-105":81,"doc-detail-272522-en":130},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":74,"head_meta":76,"extra_data":78,"updated_unix":80},105,"en","a-sat-computer-algebra-system-verification-of-the-ramsey-problems-r38-r39-master-of-mathematics-thesis","A SAT + Computer Algebra System Verification of the Ramsey Problems R(3,8) - R(3,9) - Master of Mathematics Thesis","","The Ramsey problem R(p,q) seeks the smallest n such that any graph on n vertices contains either a p-clique or an independent set of size q. This thesis provides the first certifiable proof that R(3,8)=28 by combining a Boolean satisfiability (SAT) solver with a computer algebra system (CAS). It also proves and verifies R(3,9)=36 using graph structure theory and a parallelisable SAT+CAS toolchain, extending prior non-verifiable computational results.",{"@graph":14,"@context":73},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/research-report/","Research & Report",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/a-sat-computer-algebra-system-verification-of-the-ramsey-problems-r38-r39-master-of-mathematics-thesis/272522/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/a-sat-computer-algebra-system-verification-of-the-ramsey-problems-r38-r39-master-of-mathematics-thesis/272522.png","ImageObject",300,407,{"name":42,"@type":43},"Stanford","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-31","2026-09-15",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",8,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"What does the Ramsey problem R(p,q) ask for?","Question",{"text":63,"@type":64},"It asks for the smallest n such that every graph on n vertices contains either a p-clique or an independent set of size q.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"How is R(3,8)=28 proven in this thesis?",{"text":68,"@type":64},"It is proven using a certifiable SAT+CAS combination, generating a proof automatically via SAT solving supported by computer algebra system reasoning.",{"name":70,"@type":61,"acceptedAnswer":71},"How does the approach for R(3,9)=36 differ from the R(3,8) case?",{"text":72,"@type":64},"R(3,9)=36 is established by combining structural graph theory with a parallelisable SAT+CAS toolchain, addressing prior results that were computationally solved but not verifiably proven.","https://schema.org",{"og:url":32,"og:type":75,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":77,"canonical":32},"index,follow",{"doc_id":79,"site_id":7},272522,1789465725,{"code":4,"msg":82,"data":83},"success",[84,88,92,96,101,106,111,114,119,122,126],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":85,"show_sort_weight":86,"slug":87},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":89,"show_sort_weight":90,"slug":91},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":93,"show_sort_weight":94,"slug":95},"Exam",70,"exam",{"id":97,"doc_module":4,"doc_module_name":25,"category_name":98,"show_sort_weight":99,"slug":100},5,"Comic",60,"comic",{"id":102,"doc_module":4,"doc_module_name":25,"category_name":103,"show_sort_weight":104,"slug":105},6,"Technology",50,"technology",{"id":107,"doc_module":4,"doc_module_name":25,"category_name":108,"show_sort_weight":109,"slug":110},7,"Healthcare",40,"healthcare",{"id":55,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":112,"slug":113},30,"research-report",{"id":115,"doc_module":4,"doc_module_name":25,"category_name":116,"show_sort_weight":117,"slug":118},9,"Religion & Spirituality",20,"religion-spirituality",{"id":117,"doc_module":4,"doc_module_name":25,"category_name":120,"show_sort_weight":117,"slug":121},"World Cup","world-cup",{"id":123,"doc_module":4,"doc_module_name":25,"category_name":124,"show_sort_weight":123,"slug":125},10,"Lifestyle","lifestyle",{"id":127,"doc_module":4,"doc_module_name":25,"category_name":128,"show_sort_weight":97,"slug":129},19,"General","general",{"code":4,"msg":82,"data":131},{"doc_id":79,"user_id":132,"nickname":42,"user_avatar":133,"doc_module":4,"category_id":55,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":139,"language":140,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":12,"update_tm":80,"read_time":144,"preview_page_count":4},2336477552062,"https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d","A SAT + Computer Algebra System Verification of the Ramsey Problems R(3,8)  \nand R(3,9)  \nby  \nConor Duggan  \nA thesis  \npresented to the University of Waterloo  \nin fulfillment of the  \nresearch paper requirement for the degree of  \nMaster of Mathematics  \nin  \nComputational Mathematics  \nWaterloo, Ontario, Canada, 2024  \n© Conor Duggan 2024  \nAuthor’s Declaration  \nI hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners.  \nI understand that my thesis may be made electronically available to the public.  \nAbstract  \nThe Ramsey problem R(p, q) asks for the smallest n such that every graph on n vertices must contain either a p-clique or an independent set of size q. We provide the first certifiable proof that R(3 , 8) = 28, automatically generated by a combination of Boolean satisfiability (SAT) solverand a computer algebra system (CAS) . While the R(3 , 8) problem was first computationally solved by McKay and Min in 1992, it was not a verifiable proof. This SAT+CAS combination is significantly faster than a SAT-only approach. We prove and verify R(3 , 9) = 36 combining theory on the structure of graphs and a parallelisable SAT+CAS toolchain. The theory relating to R(3 , 9) was developed by Graver and Yackel in 1968 and expanded upon and finally proved by Grinstead and Roberts in 1982, again not a verifiable proof.  \nThe SAT+CAS method that we use for our proof is very general and can be applied to a wide variety of combinatorial problems.  \nAcknowledgements  \nThe progression of mathematics and science is not a solo endeavour. My research thesis was only possible by cumulative contributions and valuable discussions.  \nForemost, I thank my supervisors Professor Vijay Ganesh and Professor Curtis Bright. Their direction and deep expertise over the course of almost two years was invaluable. Each maintain the highest standards in their work and I hope to have inherited some of their dedication to precision.  \nI owe a massive debt of gratitude to Brian Li. His support, mathematical and in other areas, was invaluable. Discussions with him are always interesting and fruitful.  \nOn a similar note, I am grateful to the contributions of Daksh Segal, Piyush Jha and Jonathon Chung.  \nFinally, to everyone who stimulated my interest in learning and in mathematics, family, friends, teachers or professors, I would not have started this journey without you.  \nDedication  \nThis is dedicated to my parents and to friends. Sometimes a walk with someone makes all the difference.  \nTable of Contents  \nAuthor’s Declaration ii  \nAbstract iii  \nAcknowledgements iv  \nDedication v  \nList of Figures viii  \nList of Tables ix  \n1 Introduction 1  \n2 Boolean Algebra, SAT Solvers and CASs 4  \n3 Cube and Conquer 6  \n4 Methodology 8  \n5 Constraints 11  \nSymmetry breaking constraints ........................... 12  \nEdge constraints ................................... 12  \nDegree constraints ................................. 14  \nTriangle constraints ................................. 14  \nMaximal clique free constraints ........................... 14  \n6 Comparison of R(3,8) Methods 16  \n7 The search for R(3,9) 18  \n8 PySMS 25  \n9 Ablation Study 27  \n10 Related work 29  \n11 Formal Verification and Result 32  \n12 Conclusion 35  \nReferences 36  \nList of Figures  \n1.1 A red/blue edge coloring on 8 vertices without a blue 3-clique or red 4-clique, showing R(3 , 4) > 8................................. 2  \n4.1 A flowchart of the orderly generation algorithm implemented as part of MathCheck’s SAT+CAS architecture........................... 9  \n4.2 Flowchart of pipeline without parallelisation from input parameters to verification. 10  \n7.1 Graph with a preferred vertex v, H1 (v) and H2 (v) . Note for p = 3, H1 (v) is an independent set.................................... 19  \n11.1 Solve time for the 10.1k cubes solved by SAT+CAS. Unsolved cubes were further cubed. Cube","cbCairn5taNKOYJl","https://ap.wps.com/l/cbCairn5taNKOYJl","pdf",363550,48,"English","# 1 Introduction\n# 2 Boolean Algebra, SAT Solvers and CASs\n# 3 Cube and Conquer\n# 4 Methodology\n## Constraints\n## Symmetry breaking constraints\n## Edge constraints\n## Degree constraints\n## Triangle constraints\n## Maximal clique free constraints\n# 6 Comparison of R(3,8) Methods\n# 7 The search for R(3,9)\n# 8 PySMS\n# 9 Ablation Study\n# 10 Related work\n# 11 Formal Verification and Result\n# 12 Conclusion","[{\"question\":\"What does the Ramsey problem R(p,q) ask for?\",\"answer\":\"It asks for the smallest n such that every graph on n vertices contains either a p-clique or an independent set of size q.\"},{\"question\":\"How is R(3,8)=28 proven in this thesis?\",\"answer\":\"It is proven using a certifiable SAT+CAS combination, generating a proof automatically via SAT solving supported by computer algebra system reasoning.\"},{\"question\":\"How does the approach for R(3,9)=36 differ from the R(3,8) case?\",\"answer\":\"R(3,9)=36 is established by combining structural graph theory with a parallelisable SAT+CAS toolchain, addressing prior results that were computationally solved but not verifiably proven.\"}]","A SAT + Computer Algebra System Verification of the Ramsey Problems R(3,8) - R(3,9) - Master of Mathematics Thesis | PDF",121]