{"paper":{"title":"Geometric duality, perfect graphs, and the Sierpi\\'nski space","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Duality between combinatorial Banach spaces holds precisely when the families are finite cliques and anti-cliques of a perfect graph on the naturals.","cross_cats":["math.CO","math.LO"],"primary_cat":"math.FA","authors_text":"Anna Pelczar-Barwacz, Barnab\\'as Farkas, Piotr Borodulin-Nadzieja","submitted_at":"2026-05-13T19:48:20Z","abstract_excerpt":"In their classical paper \\emph{On the stopping time Banach space}, Bang and Odell, among a plethora of results concerning the dyadic stopping time space and its dual, presented the first non-trivial example of the \\emph{duality phenomenon} between combinatorial Banach spaces. We give a full characterization of such pairs $(\\mc{F}_0, \\mc{F}_1)$ of families of finite sets: This duality holds iff there is a perfect graph $G$ on $\\NN$ such that $\\mc{F}_0$ consists of all finite cliques of $G$ and $\\mc{F}_1$ consists of all finite anti-cliques of $G$. As it turns out, Lov\\'asz' famous perfect graph"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"This duality holds iff there is a perfect graph G on NN such that F0 consists of all finite cliques of G and F1 consists of all finite anti-cliques of G.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The precise definition of the 'duality phenomenon' between the two combinatorial Banach spaces is taken from the Bang-Odell paper and is assumed to be the correct notion of duality for the characterization to apply.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Duality between combinatorial Banach spaces holds precisely when the families are the finite cliques and anti-cliques of a perfect graph on the naturals, making Lovász' perfect graph theorem a corollary, with further study of the Sierpiński graph case.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Duality between combinatorial Banach spaces holds precisely when the families are finite cliques and anti-cliques of a perfect graph on the naturals.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"768557cf9c193457bb686dc926e4a3899b1557ae555764e8cc5b7f78a2fa19ce"},"source":{"id":"2605.14072","kind":"arxiv","version":1},"verdict":{"id":"cd7f1369-959c-43d8-881e-a3fb5967ee8c","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T02:13:38.931996Z","strongest_claim":"This duality holds iff there is a perfect graph G on NN such that F0 consists of all finite cliques of G and F1 consists of all finite anti-cliques of G.","one_line_summary":"Duality between combinatorial Banach spaces holds precisely when the families are the finite cliques and anti-cliques of a perfect graph on the naturals, making Lovász' perfect graph theorem a corollary, with further study of the Sierpiński graph case.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The precise definition of the 'duality phenomenon' between the two combinatorial Banach spaces is taken from the Bang-Odell paper and is assumed to be the correct notion of duality for the characterization to apply.","pith_extraction_headline":"Duality between combinatorial Banach spaces holds precisely when the families are finite cliques and anti-cliques of a perfect graph on the naturals."},"references":{"count":27,"sample":[{"doi":"","year":1992,"title":"D. E. ALSPACH ANDS. ARGYROS,Complexity of weakly null sequences, vol. 321 of Dissertationes Math. (Rozprawy Mat.), 1992","work_id":"af6c33bf-a235-46a7-89e2-78a521fbaf5f","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2015,"title":"APATSIDIS,Operators on the stopping time space, Studia Math., 228 (2015), pp","work_id":"2f788b0b-5e3e-4caa-b1a6-cda700365bd6","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1972,"title":"K. A. BAKER, P. FISHBURN,ANDF. S. ROBERTS,Partial orders of dimension2, Networks, 2 (1972), pp. 11–28","work_id":"7ebc40cc-9551-41c8-a3d7-043610c5ff75","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1989,"title":"H. BANG ANDE. ODELL,On the stopping time Banach space, Quart. J. Math. Oxford Ser. (2), 40 (1989), pp. 257–273","work_id":"49659079-e18e-4640-9435-61b00710a7e5","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2019,"title":"K. BEANLAND, N. DUNCAN, M. HOLT,ANDJ. QUIGLEY,Extreme points for combinatorial Banach spaces, Glasg. Math. J., 61 (2019), pp. 487–500","work_id":"3e462502-0cd8-497b-b2c7-5fde71f215e1","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":27,"snapshot_sha256":"d29bbe866ae3b54e9fad0ab595e61c8f19c9588ddfbb3362606d4eb4dff3b23a","internal_anchors":0},"formal_canon":{"evidence_count":1,"snapshot_sha256":"d63ed92b6bf7113133509ddf09a06a7106137fd26a489221e786925072423ca0"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}