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We also prove that any set of $t=1/10n^{1/4}$ trees $T_1,T_2,..., T_t$ such that no tree is a star and $T_i$ has $n-i+1$ vertices pack into $K_{n}$ (for $n$ large enough). Finally, we prove that $t=1/4n^{1/3}$ trees $T_1,T_2,..., T_t$ such that $T_i$ has $n-i+1$ vertices pack into $K_n$ as long as each tree "},"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":"1212.3627","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-12-14T22:27:21Z","cross_cats_sorted":[],"title_canon_sha256":"4ba07dcb3c9dd5e23991019a8c4189fe08f82a91a6883c0b5edcfba64fc7836e","abstract_canon_sha256":"72933e7343456e0c0a4e42ba3e320a55bf329fdfc6a2df8b578504770827a148"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:38:22.808135Z","signature_b64":"08bz7ix0NsWZCjl70EyvswCXD/OPoUa4Uci6q6h0bUQ/9l3CYw2c0FD+qnEjYe+lOUcNccviNI+kpXPEOE2ADw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c1e79e98d9ebc5e94ff3473f6f8bff3421d46c65cdd521181e6fe51ff7be016e","last_reissued_at":"2026-05-18T03:38:22.807632Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:38:22.807632Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the tree packing conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Cory Palmer, J\\'ozsef Balogh","submitted_at":"2012-12-14T22:27:21Z","abstract_excerpt":"The Gy\\'arf\\'as tree packing conjecture states that any set of $n-1$ trees $T_{1},T_{2},..., T_{n-1}$ such that $T_i$ has $n-i+1$ vertices pack into $K_n$. We show that $t=1/10n^{1/4}$ trees $T_1,T_2,..., T_t$ such that $T_i$ has $n-i+1$ vertices pack into $K_{n+1}$ (for $n$ large enough). We also prove that any set of $t=1/10n^{1/4}$ trees $T_1,T_2,..., T_t$ such that no tree is a star and $T_i$ has $n-i+1$ vertices pack into $K_{n}$ (for $n$ large enough). 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