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In this paper, we narrow Przytycki's bounds by showing that $$ \\mathcal{N}_{k}(S) =O \\left( \\frac{ |\\chi|^{3k}}{ ( \\log |\\chi| )^2 } \\right) , $$ In particular, the size of a maximal 1-system grows sub-cubically in $|\\chi(S)|$. The proof uses a circle packing argument of Aougab-Souto and a bound for the number of curves of length at most $L$ on a hyperboli"},"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":"1610.06514","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-10-20T17:39:22Z","cross_cats_sorted":[],"title_canon_sha256":"76f0130981d4136428a122e6519e53f43657f02861d7b63fdf44d37bc0d6c970","abstract_canon_sha256":"688bf0424d5e1cfadc529fd129b7a255981f96a1b1ff3c36157b08359f4b0f8c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:01:42.996030Z","signature_b64":"rNeH4n4JNz3rv3qR6Quycl4qJf1vcD4T52+hy8gBGeL3nASvtCTLr391GnY5Zd5YwXbVREfNRf1lBDc0eIkZAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8e8d9c7bc4325db25010124f5454680a218326b2e6ce78f0d9247fb932a207a9","last_reissued_at":"2026-05-18T01:01:42.995418Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:01:42.995418Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Packing curves on surfaces with few intersections","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Ian Biringer, Jonah Gaster, Tarik Aougab","submitted_at":"2016-10-20T17:39:22Z","abstract_excerpt":"Przytycki has shown that the size $\\mathcal{N}_{k}(S)$ of a maximal collection of simple closed curves that pairwise intersect at most $k$ times on a topological surface $S$ grows at most as a polynomial in $|\\chi(S)|$ of degree $k^{2}+k+1$. In this paper, we narrow Przytycki's bounds by showing that $$ \\mathcal{N}_{k}(S) =O \\left( \\frac{ |\\chi|^{3k}}{ ( \\log |\\chi| )^2 } \\right) , $$ In particular, the size of a maximal 1-system grows sub-cubically in $|\\chi(S)|$. 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