{"paper":{"title":"A non-hereditary Pollyanna class that is not strongly Pollyanna","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"A non-hereditary graph class exists that is Pollyanna but fails to be strongly Pollyanna for every k.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hongzhang Chen, Kaiyang Lan","submitted_at":"2026-05-14T08:26:12Z","abstract_excerpt":"Chudnovsky, Cook, Davies, and Oum introduced the notion of Pollyanna graph classes: a class $\\mathcal{C}$ is Pollyanna if for every $\\chi$-bounded class $\\mathcal{F}$, the intersection $\\mathcal{C} \\cap \\mathcal{F}$ is polynomially $\\chi$-bounded. They further defined $\\mathcal{C}$ to be strongly Pollyanna if it is $k$-strongly Pollyanna for some integer $k$, meaning that $\\mathcal{C} \\cap \\mathcal{F}$ is polynomially $\\chi$-bounded for every $k$-good class $\\mathcal{F}$. They asked whether there are Pollyanna graph classes that are not strongly Pollyanna. In this note we answer this question "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We construct a class C that is Pollyanna but, for every k ≥ 1, is not k-strongly Pollyanna; in particular C is not strongly Pollyanna.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"Graph classes are not required to be hereditary, allowing the construction of a non-hereditary class C that separates the Pollyanna and strongly Pollyanna properties.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A non-hereditary graph class exists that is Pollyanna but not strongly Pollyanna.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A non-hereditary graph class exists that is Pollyanna but fails to be strongly Pollyanna for every k.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"be1069629c657579fa8a33693914e1876a759f121d11a05ef92a37b54e0e6f9d"},"source":{"id":"2605.14547","kind":"arxiv","version":1},"verdict":{"id":"288cd3e9-a257-468d-bed2-05a56aaed374","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T01:23:29.888026Z","strongest_claim":"We construct a class C that is Pollyanna but, for every k ≥ 1, is not k-strongly Pollyanna; in particular C is not strongly Pollyanna.","one_line_summary":"A non-hereditary graph class exists that is Pollyanna but not strongly Pollyanna.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"Graph classes are not required to be hereditary, allowing the construction of a non-hereditary class C that separates the Pollyanna and strongly Pollyanna properties.","pith_extraction_headline":"A non-hereditary graph class exists that is Pollyanna but fails to be strongly Pollyanna for every k."},"references":{"count":7,"sample":[{"doi":"","year":2024,"title":"M. Bria´ nski, J. Davies and B. Walczak, Separating polynomialχ-boundedness fromχ- boundedness,Combinatorica44(2024), 1–8","work_id":"0c738cc1-1933-43f0-bd4f-c329505f5503","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2026,"title":"M. Chudnovsky, L. Cook, J. Davies and S. Oum, Reunitingχ-boundedness with polynomial χ-boundedness,J. Combin. Theory Ser. B176(2026), 30–73. 6","work_id":"33eb6729-c40d-447a-8eb0-708ccee5d046","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2017,"title":"Esperet, Graph colorings, flows and perfect matchings, Habilitation Thesis, Universit´ e Grenoble Alpes, 2017","work_id":"a13d5e44-c462-41c3-a4c6-a1090a8373d0","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1973,"title":"Gy´ arfas, On Ramsey covering-numbers, in:Infinite and Finite Sets (Colloq., Keszthely, 1973; Dedicated to P","work_id":"34cdf834-856f-48f9-9f1d-7b9f8dcac34c","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1959,"title":"Erd˝ os, Graph theory and probability,Canad","work_id":"f15e8ca4-0a0f-400a-bc62-66d9d4925b2a","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":7,"snapshot_sha256":"f77cd6ff95b98e7f4c7aee9e739aa7b62588e5736e97b1f2c74dedfa20c7ab21","internal_anchors":0},"formal_canon":{"evidence_count":1,"snapshot_sha256":"e614044c597dac5eebe73693da8fa38730294ecfa0d427d9d25948a134822316"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}