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Construct a graph, called the distance graph of $E$, by letting the vertices be the elements of $E$ and connect a pair of vertices corresponding to vectors $x,y \\in E$ by an edge if $||x-y||={(x_1-y_1)}^2+\\dots+{(x_d-y_d)}^2=1$. We shall prove that if the size of $E$ is sufficiently large, then the distance graph of $E$ contains long non-overlapping paths and vertices of high degree."},"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":"1406.0107","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-05-31T20:09:34Z","cross_cats_sorted":["math.CA","math.NT"],"title_canon_sha256":"6e10986553e3e52106aa266d529f12142878f6f60c651d53f3caf01fbcff2915","abstract_canon_sha256":"9d28b69f51fbbe28c5ee2664f8d10c2c622552069420f2b5393523dc825029a6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:50:41.760132Z","signature_b64":"rScFHJJGEncnEzm5CFM2SmQOepLO8w88+WYOtgh5UrUML22/K0E+jWCUr4k66DzETXINWXrGxnPvwRhm+UM8Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c09798596119b487ae5fc0e4e950e72afc4588a86ba3f4d76c3258654a776aba","last_reissued_at":"2026-05-18T02:50:41.759680Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:50:41.759680Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Long paths in the distance graph over large subsets of vector spaces over finite fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.NT"],"primary_cat":"math.CO","authors_text":"A. 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