{"paper":{"title":"On Tournament Anti-Sidorenko Orientations of Trees","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Oriented paths with exactly one non-leaf source or sink are tournament anti-Sidorenko.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Felix Christian Clemen, Hao Chen, Jonathan A. Noel","submitted_at":"2026-05-13T21:44:25Z","abstract_excerpt":"An oriented graph $\\vec{H}$ is said to be tournament anti-Sidorenko if the homomorphism density of $\\vec{H}$ in any tournament $\\vec{T}$ is bounded above by the homomorphism density of $\\vec{H}$ in a large uniformly random tournament. We prove the following:\n  (1) Every oriented path with at least three arcs and exactly one non-leaf source or sink vertex is tournament anti-Sidorenko.\n  (2) An oriented path is tournament anti-Sidorenko if the distance between any leaf vertex and any source or sink vertex is at least two and the distance between any pair of non-leaf source or sink vertices is a "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Every oriented path with at least three arcs and exactly one non-leaf source or sink vertex is tournament anti-Sidorenko. An oriented path is tournament anti-Sidorenko if the distance between any leaf vertex and any source or sink vertex is at least two and the distance between any pair of non-leaf source or sink vertices is a multiple of four. Every spider with exactly three legs admits a tournament anti-Sidorenko orientation.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The proofs for the stated distance conditions and structural properties of the oriented paths and spiders hold without gaps or unstated restrictions on the graphs considered.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Specific orientations of paths with distance conditions and three-legged spiders are tournament anti-Sidorenko, proving conjectures and yielding new families.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Oriented paths with exactly one non-leaf source or sink are tournament anti-Sidorenko.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"e425063248ef346e7843a4c7e4a3ad5e6eaf74445beaeadad3593d732c0649b0"},"source":{"id":"2605.14138","kind":"arxiv","version":1},"verdict":{"id":"23160e29-80d9-4091-b7e1-eeb59c8e59f7","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T02:16:07.669349Z","strongest_claim":"Every oriented path with at least three arcs and exactly one non-leaf source or sink vertex is tournament anti-Sidorenko. An oriented path is tournament anti-Sidorenko if the distance between any leaf vertex and any source or sink vertex is at least two and the distance between any pair of non-leaf source or sink vertices is a multiple of four. Every spider with exactly three legs admits a tournament anti-Sidorenko orientation.","one_line_summary":"Specific orientations of paths with distance conditions and three-legged spiders are tournament anti-Sidorenko, proving conjectures and yielding new families.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The proofs for the stated distance conditions and structural properties of the oriented paths and spiders hold without gaps or unstated restrictions on the graphs considered.","pith_extraction_headline":"Oriented paths with exactly one non-leaf source or sink are tournament anti-Sidorenko."},"references":{"count":30,"sample":[{"doi":"","year":2025,"title":"A. Basit, B. Granet, D. Horsley, A. K¨ undgen, and K. Staden. The semi-inducibility problem. E-print arXiv:2501.09842v2, 2025","work_id":"9637b173-9e80-419a-ba4e-ee914a5490cf","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2025,"title":"N. Behague, G. Crudele, J. A. Noel, and L. M. Simbaqueba. Sidorenko-Type Inequalities for Pairs of Trees.Random Structures Algorithms, 67(1):Paper No. e70026, 2025","work_id":"22515211-e0d0-48a8-862f-6e841cc502ad","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"N. Behague, N. Morrison, and J. A. Noel. Off-diagonal commonality of graphs via entropy.SIAM J. Discrete Math., 38(3):2335–2360, 2024","work_id":"39d76ff8-1c81-44f4-872f-617dd8c6730c","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2026,"title":"V. Bitonti, E. Hogan, J. A. Noel, and D. Tsarev. Relative Sidorenko inequalities in oriented graphs. In preparation, 2026","work_id":"408cb517-1458-4d27-bd5a-7cc07028aa73","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2022,"title":"G. Blekherman and A. Raymond. A path forward: tropicalization in extremal combi- natorics.Adv. Math., 407:Paper No. 108561, 68, 2022","work_id":"9f7ca14d-60f7-4897-95d5-935c98cbcc48","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":30,"snapshot_sha256":"56d637ad18a24d6980082a179058ca11a30ac23253912f425adfc9c40fc69b53","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}