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We show the following.\n  For every integer $d\\geq 1$, every $b_1,...,b_d\\in\\mathbb{N}$ with $b_i\\geq 2$ for all $i\\in\\{1,...,d\\}$, every integer $k\\meg 1$ and every real $0<\\epsilon\\leq 1$ there exists an integer $N$ with the following property. If $(T_1,...,T_d)$ are homogeneous trees such that the branchin"},"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":"1105.2419","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-05-12T10:47:38Z","cross_cats_sorted":[],"title_canon_sha256":"8370fda16c56a71b243dc04a54856de6e381f8e638f8cf374a9b76a7aa1c1716","abstract_canon_sha256":"2aa955108ca6ca08cefe7df26120c8d77ee2aca7bdba651810ae62cce4d0efb8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:30:28.308701Z","signature_b64":"kZoZOZMuRRwilWaAMBUYBN1hbKM9s6CAQb5JH6wenPHG+x1dq5DD3tl8ZBxTz8oauMH2lDHAwbwZ02TOiPodBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ca68b02755226aaceda8859354b276acdc67b5691c48575525233dc6a67cfc5e","last_reissued_at":"2026-05-18T03:30:28.307994Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:30:28.307994Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Dense subsets of products of finite trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Konstantinos Tyros, Pandelis Dodos, Vassilis Kanellopoulos","submitted_at":"2011-05-12T10:47:38Z","abstract_excerpt":"We prove a \"uniform\" version of the finite density Halpern-L\\\"{a}uchli Theorem. Specifically, we say that a tree $T$ is homogeneous if it is uniquely rooted and there is an integer $b\\geq 2$, called the branching number of $T$, such that every $t\\in T$ has exactly $b$ immediate successors. We show the following.\n  For every integer $d\\geq 1$, every $b_1,...,b_d\\in\\mathbb{N}$ with $b_i\\geq 2$ for all $i\\in\\{1,...,d\\}$, every integer $k\\meg 1$ and every real $0<\\epsilon\\leq 1$ there exists an integer $N$ with the following property. 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