[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83147-en":3,"doc-seo-83147-105":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":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"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},83147,687197207057,"Sage","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Computation of small reflective and dihedral Ramsey numbers","Reflective and dihedral Ramsey numbers are treated as a specialized family of permutational Ramsey numbers defined via graph embeddings compatible with permutation groups on ordered vertex sets. Using the SAT-based strategy originally developed by Poljak for ordered Ramsey numbers and extended to cyclic settings, exact values and lower bounds are computed for small two-color reflective and dihedral Ramsey numbers with arguments from monotone/alternating paths, monotone cycles, start-central stars, complete graphs, and nested matchings. General results and conjectures are derived from the computational outcomes.","arXiv :2607 .06817v2 [math .CO] 12 Jul 2026  \nComputation of small reflective and dihedral Ramsey numbers  \nIvan Damnjanović∗1,2 and Irena Ðorđević 1  \n1 Faculty of Electronic Engineering, University of Niš, Aleksandra Medvedeva 4, Niš, 18104, Serbia  \n2 Faculty of Mathematics, Natural Sciences and Information Technologies, University of Primorska,  \nGlagoljaška 8, Koper, 6000, Slovenia  \nAbstract  \nThroughout, all graphs are simple, finite and have vertex sets of the form {0, 1 , 2 , . . . , n − 1} for some n ∈ N. For graphs G and H , and a permutation group Γ on the vertex set of H , we say that H is Γ -embeddable in G if there exists a graph homomorphism from H to G of the form ψ◦φ, where φ ∈ Γ and ψ is an increasing injection. Recently, standard and ordered Ramsey numbers of graphs were unified through the introduction of permutational Ramsey numbers, defined as follows. For graphs H1 , H2 , . . . , Hk and permutation groups Γ1 , Γ2 ,   , Γk on their respective vertex sets, the permutational Ramsey number R (HΓ11 , HΓ22 ,   , HΓkk ) is the minimum n ∈ N such that for every k-edge-coloring of a complete graph on n vertices, there exists some j ∈ {1, 2 ,..., k} for which Hj is Γj-embeddable in the spanning subgraph of the complete graph comprising the edges of color j.  \nHere, we consider reflective (resp. dihedral) Ramsey numbers, which are a specific class of permutational Ramsey numbers in which each group Γj is the reflection group (resp. dihedral group) on the naturally ordered vertex set of Hj . Focusing on the two-color case, we apply the SAT-based approach originally proposed by Poljak for ordered Ramsey numbers and recently extended to cyclic Ramsey numbers. We utilize the Kissat SAT solver to obtain exact values and lower bounds for small reflective and dihedral Ramsey numbers whose two arguments belong to the following graph classes: monotone and alternating paths, monotone cycles, start-central stars, complete graphs and nested matchings. We also derive several general results and formulate conjectures based on the computational findings.  \nKeywords: Ramsey numbers, reflective Ramsey numbers, dihedral Ramsey numbers, permutational Ramsey numbers, SAT.  \nMathematics Subject Classification (2020): 05D10, 05C55 .  \n1 Introduction  \nWe consider all graphs to be undirected, simple and finite, with vertex sets of the form {0, 1 , 2 , . . . , n − 1} for some n ∈ N. For a graph G, we denote its vertex and edge sets by V (G) and E (G), respectively, and write |G| for its order. We write Kn for the complete graph on n vertices. A k-edge-coloring of a graph G is a mapping from E (G) to {1, 2 ,..., k} .  \nRamsey theory originates from a theorem of Ramsey [25], which asserts that for any graphs H1 , H2 ,..., Hk , every k-edge-coloring of Kn contains a monochromatic copy of Hj in color j, for some j ∈ {1, 2 ,..., k}, provided that n ∈ N is sufficiently large. The minimum such n is called the Ramsey number R (H1 , H2 ,..., Hk) . Although the definition of Ramsey numbers is straightforward, determining their exact values is often remarkably difficult. Indeed, even the diagonal Ramsey number R (K5 , K5 ) remains unknown [24] . Ramsey-type phenomena have been investigated in numerous settings over the past century, leading to a rich literature on a variety of combinatorial structures, including hypergraphs, ordered graphs, geometric configurations, and sequences; see [11, 13–15, 17 , 20–22, 24] and the references therein.  \nAmong the many directions in modern Ramsey theory, ordered Ramsey numbers have received considerable attention in recent years. Given graphs H1 , H2 ,..., Hk , the ordered Ramsey number Rord (H1 , H2 ,..., Hk) is defined as the minimum n ∈ N such that for every k-edge-coloring of Kn , there exists an increasing injective homomorphism from Hj to the spanning subgraph of Kn comprising the edges of color j, for some j ∈ {1, 2 ,..., k} . The systematic study of ordered Ramsey numbers was initiated independently by ","cbCaifw0bZZQ7DvD","https://ap.wps.com/l/cbCaifw0bZZQ7DvD","pdf",634202,1,17,"English","en",105,"# Abstract\n# Introduction\n## Ramsey theory background\n## Ordered and permutational Ramsey numbers\n## Reflective and dihedral Ramsey numbers and computational approach","[{\"question\":\"What are permutational Ramsey numbers in the document?\",\"answer\":\"They are defined as the minimum n such that every k-edge-coloring of a complete graph on n vertices yields, for some color j, an embedding of Hj into the color-j spanning subgraph that respects a specified permutation group Γj.\"},{\"question\":\"How are reflective and dihedral Ramsey numbers obtained from the permutational framework?\",\"answer\":\"For each Hj, the permutation group Γj is chosen to be the reflection group or, respectively, the dihedral group acting on the naturally ordered vertex set of Hj.\"},{\"question\":\"Which method and SAT solver are used to compute small two-color reflective and dihedral Ramsey numbers?\",\"answer\":\"The work uses an SAT-based approach (originating with Poljak for ordered Ramsey numbers and extended to cyclic Ramsey numbers) implemented with the Kissat SAT solver to obtain exact values and lower bounds.\"}]",1784185604,43,{"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},"computation-of-small-reflective-and-dihedral-ramsey-numbers","",{"@graph":35,"@context":84},[36,53,67],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/computation-of-small-reflective-and-dihedral-ramsey-numbers/83147/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What are permutational Ramsey numbers in the document?","Question",{"text":74,"@type":75},"They are defined as the minimum n such that every k-edge-coloring of a complete graph on n vertices yields, for some color j, an embedding of Hj into the color-j spanning subgraph that respects a specified permutation group Γj.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How are reflective and dihedral Ramsey numbers obtained from the permutational framework?",{"text":79,"@type":75},"For each Hj, the permutation group Γj is chosen to be the reflection group or, respectively, the dihedral group acting on the naturally ordered vertex set of Hj.",{"name":81,"@type":72,"acceptedAnswer":82},"Which method and SAT solver are used to compute small two-color reflective and dihedral Ramsey numbers?",{"text":83,"@type":75},"The work uses an SAT-based approach (originating with Poljak for ordered Ramsey numbers and extended to cyclic Ramsey numbers) implemented with the Kissat SAT solver to obtain exact values and lower bounds.","https://schema.org",{"og:url":51,"og:type":86,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":88,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":91},[92,96,100,104,109,114,119,122,127,130,134],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":93,"show_sort_weight":94,"slug":95},"Story & 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