[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85440-en":3,"doc-seo-85440-105":29,"detail-sidebar-cat-0-en-105":90},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":11,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},85440,687197100911,"Himbo","https://ap-avatar.wpscdn.com/avatar/a000239b6f1da00475?x-image-process=image/resize,m_fixed,w_180,h_180&k=1782698725881665579",8,"Research & Report","Constraint Satisfaction Problems, Compactness and Non-Measurable Sets","A finite relational structure A is compact when, for any infinite relational structure B of the same type, A admits a homomorphism from B exactly when it admits homomorphisms from every finite substructure of B. The work proves that compactness of A can be established in Zermelo–Fraenkel set theory when A has width 1. Otherwise, assuming compactness forces the existence of non-Lebesgue-measurable sets in three-dimensional space, linking CSP complexity to set-theoretic strength.","Logical Methods in Computer Science Volume 22, Issue 3, 2026, pp. 1:1–1:8  \n[https://lmcs.episciences.org/](https://lmcs.episciences.org/)  \nSubmitted Published  \nAug. 21, 2025 Jul. 14, 2026  \nCONSTRAINT SATISFACTION PROBLEMS, COMPACTNESS AND  \nNON-MEASURABLE SETS  \nCLAUDE TARDIF   \nDepartment of Mathematics and Computer Science, Royal Military College of Canada, Kingston, Ontario, Canada.  \ne-mail address: [Claude.Tardif@rmc.ca](Claude.Tardif@rmc.ca)  \nAbstract. A finite relational structure A is called compact if for any infinite relational structure B of the same type, the existence of a homomorphism from B to A is equivalent to the existence of homomorphisms from all finite substructures of B to A. We show that if A has width 1, then the compactness of A can be proved in the axiom system of Zermelo and Fraenkel, but otherwise, the compactness of A implies the existence of non-measurable sets in 3-space.  \n1. Introduction  \nA graph is bipartite if and only if it does not contain an odd cycle. Already in 1916, K¨onig [K¨on16] acknowledged that this theorem requires the axiom of choice for the infinite case. One of the two possible colourings must be selected for each connected component. Later in 1961, Mycielski [Myc61] proved that in the infinite case, K¨onig’s theorem is precisely equivalent to the axiom of choice for sets of pairs. Now, consider the statement that adigraph admits a homomorphism to the transitive tournament T2 on two vertices if and only if it admits no homomorphism from the directed path P2 with two consecutive arcs. Even in the infinite case, it requires no version of the axiom of choice. Indeed, if there are no homomorphic images of P2 , every vertex is a source or a sink, possibly both, and we can define a homomorphism to T2 by mapping all sources to the source of T2 and all other vertices to the sink.  \nA finite relational structure A is called compact if the existence of a homomorphism from a structure B to A is equivalent to the existence of homomorphisms from all finite substructures of B to A. The axiom of choice implies the all-encompassing compactness theorem, which implies that every finite relational structure is compact. More precisely, the compactness theorem is equivalent to the ultrafilter axiom, which states that any filter on a set can be extended to an ultrafilter. It is weaker than the axiom of choice. In graph theory, the compactness of the complete graphs is called the de Bruijn-Erd˝os theorem, after the 1951 result of de Bruijn and Erd˝os [dBE51] . Later in 1971, L¨auchli [L¨au71] proved that for any n ≥ 3, the compactness of the complete graph Kn already implies the ultrafilter axiom. Thus, viewed as an axiom, the statement “K3 is compact” is stronger than “K2 is compact”, and it is at least as strong as “A is compact” for any finite relational structure A.  \nKey words and phrases: Constraint satisfaction problems, Zermelo and Fraenkel set theory, non-measurable sets.  \nl LOGICAL METHODS © COMPACTNESS AND NON-MEASURABLE SETS  \nIN COMPUTER SCIENCE DOI:10 .46298/LMCS-22(3:1)2026 ⃝CC Creative Commons  \nRecently, the following generalisation of L¨auchli’s result was obtained by K´atay, T´oth, and Vidny´anszky.  \nTheorem 1.1 [KTV24] . Let A be a finite relational structure. Then the compactness of A implies the ultrafilter axiom if and only if A does not admit a cyclic polymorphism.  \nThe “cyclic polymorphisms” involved will be defined in Section 2 . They provide the criterion separating the simple from the complex in the dichotomy of constraint satisfaction problems. For a given relational structure A, the corresponding constraint satisfaction problem CSP(A) is the problem of deciding whether an input structure B admits a homomorphism to A. Bulatov [Bul17] and Zhuk [Zhu20] independently proved that CSP (A) is polynomial if A admits a cyclic polymorphism, and NP-complete otherwise. This dichotomy between tractable and intractable constraint satisfaction problems is the one that had been","cbCaijymKW4KFAdR","https://ap.wps.com/l/cbCaijymKW4KFAdR","pdf",349966,2,1,"English","en",105,"# Introduction\n## Compactness and axioms of set theory\n## CSP dichotomy via cyclic polymorphisms\n## Width 1 and tree duality\n## Main implication: non-measurable sets","[{\"question\":\"What does it mean for a finite relational structure to be compact?\",\"answer\":\"A is compact if, for any infinite relational structure B of the same type, homomorphisms from B to A exist exactly when homomorphisms exist from all finite substructures of B to A.\"},{\"question\":\"How does width 1 affect the set-theoretic provability of compactness?\",\"answer\":\"If A has width 1, compactness of A can be proved within the Zermelo–Fraenkel axiom system.\"},{\"question\":\"What happens if A does not have width 1?\",\"answer\":\"If A lacks width 1, then the axiom “A is compact” implies the existence of a set in R3 that is not Lebesgue measurable.\"}]",1784203538,20,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":85,"head_meta":87,"extra_data":89,"updated_unix":27},"constraint-satisfaction-problems-compactness-and-non-measurable-sets","",{"@graph":35,"@context":84},[36,52,67],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,46,49],{"item":40,"name":41,"@type":42,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":20},"https://docshare.wps.com/document/","Document",{"item":47,"name":12,"@type":42,"position":48},"https://docshare.wps.com/document/research-report/",3,{"item":50,"name":13,"@type":42,"position":51},"https://docshare.wps.com/document/constraint-satisfaction-problems-compactness-and-non-measurable-sets/85440/",4,{"url":50,"name":13,"@type":53,"author":54,"headline":13,"publisher":56,"fileFormat":59,"inLanguage":23,"description":14,"dateModified":60,"datePublished":61,"encodingFormat":59,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":55},"Person",{"url":40,"name":57,"@type":58},"DocShare","Organization","application/pdf","2026-07-21","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":20},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What does it mean for a finite relational structure to be compact?","Question",{"text":74,"@type":75},"A is compact if, for any infinite relational structure B of the same type, homomorphisms from B to A exist exactly when homomorphisms exist from all finite substructures of B to A.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How does width 1 affect the set-theoretic provability of compactness?",{"text":79,"@type":75},"If A has width 1, compactness of A can be proved within the Zermelo–Fraenkel axiom system.",{"name":81,"@type":72,"acceptedAnswer":82},"What happens if A does not have width 1?",{"text":83,"@type":75},"If A lacks width 1, then the axiom “A is compact” implies the existence of a set in R3 that is not Lebesgue measurable.","https://schema.org",{"og:url":50,"og:type":86,"og:title":13,"og:site_name":57,"og:description":14},"article",{"robots":88,"canonical":50},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":91},[92,96,100,104,109,114,119,122,126,129,133],{"id":21,"doc_module":4,"doc_module_name":45,"category_name":93,"show_sort_weight":94,"slug":95},"Story & Novel",90,"story-novel",{"id":20,"doc_module":4,"doc_module_name":45,"category_name":97,"show_sort_weight":98,"slug":99},"Literature",80,"literature",{"id":51,"doc_module":4,"doc_module_name":45,"category_name":101,"show_sort_weight":102,"slug":103},"Exam",70,"exam",{"id":105,"doc_module":4,"doc_module_name":45,"category_name":106,"show_sort_weight":107,"slug":108},5,"Comic",60,"comic",{"id":110,"doc_module":4,"doc_module_name":45,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":45,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":45,"category_name":124,"show_sort_weight":28,"slug":125},9,"Religion & Spirituality","religion-spirituality",{"id":28,"doc_module":4,"doc_module_name":45,"category_name":127,"show_sort_weight":28,"slug":128},"World Cup","world-cup",{"id":130,"doc_module":4,"doc_module_name":45,"category_name":131,"show_sort_weight":130,"slug":132},10,"Lifestyle","lifestyle",{"id":134,"doc_module":4,"doc_module_name":45,"category_name":135,"show_sort_weight":105,"slug":136},19,"General","general"]