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Let $\\cF \\subseteq {[n] \\choose d}$ be an arbitrary $d$-uniform set system such that $\\cF$ does not shatter an $s+1$-element set, then $$ |\\cF|\\leq {n \\choose s}.$$ We prove here two generalizations of the above theorem to $n$-tuple systems. To obtain these results, we use Gr\\\"obner basis methods, and describe the standard monomials of Hamming spheres."},"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":"1008.4660","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-08-27T08:41:00Z","cross_cats_sorted":["math.AC"],"title_canon_sha256":"7b949ba425943a8d2719212539f9b77bf89b8f71ed83fbe6b652b84eebf7073f","abstract_canon_sha256":"38e519f22df56229a6a3cc4213a50295466df8a6526a6a33ff97b59209d23284"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:02:59.672813Z","signature_b64":"Bmo9zzCrWlBRPwVOS8rGzbvXCqChCov64csCdv5RqV6kC7U+xu8O1vWh+Gs7I25xOPASskp0QUSCxsQoA7qzDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c43e2e82b7798f7667bf1e5d0ae65ba934b52c31a88ecde33891b15a9520d32b","last_reissued_at":"2026-05-18T01:02:59.672292Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:02:59.672292Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Multivalued generalizations of the Frankl--Pach Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.CO","authors_text":"Gabor Heged\\\"us, Lajos Ronyai","submitted_at":"2010-08-27T08:41:00Z","abstract_excerpt":"P. Frankl and J. Pach proved the following uniform version of Sauer's Lemma. Let $n,d,s$ be natural numbers such that $d\\leq n$, $s+1\\leq n/2$. Let $\\cF \\subseteq {[n] \\choose d}$ be an arbitrary $d$-uniform set system such that $\\cF$ does not shatter an $s+1$-element set, then $$ |\\cF|\\leq {n \\choose s}.$$ We prove here two generalizations of the above theorem to $n$-tuple systems. 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