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Dover proved that semiovals (semiarcs with $t=1$) containing $q$ collinear points exist in $\\Pi_q$ only if $q<3$. We show that if $t>1$, then $t$-semiarcs with $q+1-t$ collinear points exist only if $t\\geq \\sqrt{q-1}$. In $\\mathrm{PG}(2,q)$ we prove the lower bound $t\\geq(q-1)/2$, with equality only if ${\\ca"},"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":"1310.7204","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-10-27T15:23:36Z","cross_cats_sorted":[],"title_canon_sha256":"d08698f17401bfe8bc213c4feef10ca82a7c10ed3391716eff8f993c6546f41b","abstract_canon_sha256":"90d064fcda59f8108be477e6ca89dfaf56b3695e3d3d99d3dfdaa77265bb2f83"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:08:45.967378Z","signature_b64":"TbNM5MBLokC28R4uFlUpsYQcuB/i0ryI5NceAydcWOulP5upDCsHX/4XxwuR3m08HbwUww59+qVU5EciIWYJCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d25d564c7903e9423f449a2549a286bb369d26b61b4e655347f58d861bfeb753","last_reissued_at":"2026-05-18T03:08:45.966652Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:08:45.966652Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Semiarcs with long secants","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bence Csajb\\'ok","submitted_at":"2013-10-27T15:23:36Z","abstract_excerpt":"In a projective plane $\\Pi_q$ of order $q$, a non-empty point set ${\\cal S}_t$ is a $t$-semiarc if the number of tangent lines to ${\\cal S}_t$ at each of its points is $t$. 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