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Let $f(n,C_l)$ be the largest number of edges in a subgraph of a hypercube $Q_n$ containing no cycle of length $l$. It is known that $f(n, C_l) = o(|E(Q_n)|)$, when $l= 4k$, $k\\geq 2$ and that $f(n, C_6) \\geq \\frac{1}{3} |E(Q_n)|$. It is an open question to determine $f(n, C_l)$ for $l=4k+2$, $k\\geq 2$. Here, we give a general upper bound for $f(n,C"},"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":"1605.06572","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-05-21T02:14:11Z","cross_cats_sorted":[],"title_canon_sha256":"7ac7e536338c7a50d51e660d5161d50494a11066b75a923c950d0846c33090cf","abstract_canon_sha256":"48be271a7d9352ca25ad76e58f38c84561cb6b16fb7e5a6e9ceec95f3ff9693d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:13:46.762078Z","signature_b64":"gEO6QItWGeIx0Q3yBtOTqaVkbFnv2z3TLAqc5Z9yqi30EGfqyrhM9dZguXJftnQRVxLs+DCiNjzEEj3epcZlAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6b05738c56b93ba76a64d39f545520ad47a25f97767d64c8d17cd329fac28634","last_reissued_at":"2026-05-18T01:13:46.761415Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:13:46.761415Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A note on short cycles in a hypercube","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Maria Axenovich, Ryan R. Martin","submitted_at":"2016-05-21T02:14:11Z","abstract_excerpt":"How many edges can a quadrilateral-free subgraph of a hypercube have? This question was raised by Paul Erd\\H{o}s about $27$ years ago. His conjecture that such a subgraph asymptotically has at most half the edges of a hypercube is still unresolved. Let $f(n,C_l)$ be the largest number of edges in a subgraph of a hypercube $Q_n$ containing no cycle of length $l$. It is known that $f(n, C_l) = o(|E(Q_n)|)$, when $l= 4k$, $k\\geq 2$ and that $f(n, C_6) \\geq \\frac{1}{3} |E(Q_n)|$. It is an open question to determine $f(n, C_l)$ for $l=4k+2$, $k\\geq 2$. 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