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We prove that there exist real numbers $\\alpha, \\epsilon > 0$ such that, for sufficiently large $n$ and for every tree $T$ on $n$ vertices with maximum degree at most $n^{\\epsilon}$, Maker has a winning strategy for the $(1 : q)$ game ${\\mathcal T}_n$, for every $q \\leq n^{\\alpha}$. Moreover, we prove that Maker"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1010.2857","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-10-14T08:58:36Z","cross_cats_sorted":[],"title_canon_sha256":"32277d7560b0043629362a4286c6164e119f875f98eeacd54923e1a7e7fb26dc","abstract_canon_sha256":"f3788f4c38b84779774d9fce03d122d5166a35cc3b12bf070f3cc23fc5a9e3ba"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:39:18.998898Z","signature_b64":"Ai6Edxa6hEPxNKp9t2K87S43tVBpnVGRTODvFoX1jD5k2GK0AlRVN4Lw98Q62dbm2AazlK6eYreqSTV36SHwBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0f5dc00ff2f3be6f88df3d8d303d9341e41a7b3d0035a532ed867921469f04ae","last_reissued_at":"2026-05-18T04:39:18.998426Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:39:18.998426Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Fast embedding of spanning trees in biased Maker-Breaker games","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Asaf Ferber, Dan Hefetz, Michael Krivelevich","submitted_at":"2010-10-14T08:58:36Z","abstract_excerpt":"Given a tree $T=(V,E)$ on $n$ vertices, we consider the $(1 : q)$ Maker-Breaker tree embedding game ${\\mathcal T}_n$. 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