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If $r \\leq c \\frac{n}{\\ln n}$ then all bounds have a type $A_1(n, \\ln n, r)(\\frac{r}{r-1})^n \\leq p(n, r) \\leq A_2(n, r, \\ln r) (\\frac{r}{r-1})^n$, where $A_1$, $A_2$ are some algebraic fractions. The main result is a new lower bound on $p(n,r)$ when $r$ is at least $c \\sqrt n$; we improve an upper bound on $p(n,r)$ if $n = o(r^{3/2})$.\n  Also we show that $p(n,r)$ has upper and lower bounds depend o"},"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":"1705.03797","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-05-10T14:38:18Z","cross_cats_sorted":[],"title_canon_sha256":"23b9def4e2557f474dec4e5041cafc0ce3fe4e179aa4668f8a9e9c68fcd43e70","abstract_canon_sha256":"b0415c3ed0393d490cd45dcb1945cffecf1be7a62070e7fd7294a93834fb7a7e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:44:43.023578Z","signature_b64":"qakr6TqPCKdLpfgMgd/hxdjobCiwzxTsWrSps7WYXpz0qBOYyQoZQtR2rXWRoO+6PagHVYWlQnwrEDtQWrxGBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bc70602075baaba32e502ffd55bbe2cf24e1c2d5e9904c3aea74ef37e5b0ff22","last_reissued_at":"2026-05-18T00:44:43.023126Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:44:43.023126Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A note on panchromatic colorings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Danila Cherkashin","submitted_at":"2017-05-10T14:38:18Z","abstract_excerpt":"This paper studies the quantity $p(n,r)$, that is the minimal number of edges of an $n$-uniform hypergraph without panchromatic coloring (it means that every edge meets every color) in $r$ colors. If $r \\leq c \\frac{n}{\\ln n}$ then all bounds have a type $A_1(n, \\ln n, r)(\\frac{r}{r-1})^n \\leq p(n, r) \\leq A_2(n, r, \\ln r) (\\frac{r}{r-1})^n$, where $A_1$, $A_2$ are some algebraic fractions. 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