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The case $s=0$, which corresponds to empty monochromatic triangles, has been studied extensively over the last few years. In particular, it is known that $M_3(1, 0)=3$, $M_3(2, 0)=9$ and $M_3(c, 0)=\\infty$, for $c\\geq 3$. In this paper we extend these results when $c \\geq 2$ and $s \\geq 1$. We prove that the least integer $\\lambda_3(c)$ such tha"},"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":"1410.0424","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-10-02T00:42:50Z","cross_cats_sorted":[],"title_canon_sha256":"23da4c9c9f72ed416243d66e1410a7d72a028277fbee1978879f7cdd363da7e0","abstract_canon_sha256":"acf747e46169368fff2174df2b247c57689bd8ebbb4b8eca1c38c2b0bf140a44"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:44:11.762055Z","signature_b64":"aoNyKXgvAno6k8ks7jCZl6zeWlNK8E5f1fnvoU5R7bNEM75EVBh2ztqZXQK3mpWVe1Cazur2UVxvh+Pg9bDCCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7ca3c3527fcc11050d6fa2ae5dbc386330c242acfcc1eccf9c44bc67672ed06c","last_reissued_at":"2026-05-18T01:44:11.761371Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:44:11.761371Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Almost Empty Monochromatic Triangles in Planar Point Sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bhaswar B. Bhattacharya, Deepan Basu, Kinjal Basu, Sandip Das","submitted_at":"2014-10-02T00:42:50Z","abstract_excerpt":"For positive integers $c, s \\geq 1$, let $M_3(c, s)$ be the least integer such that any set of at least $M_3(c, s)$ points in the plane, no three on a line and colored with $c$ colors, contains a monochromatic triangle with at most $s$ interior points. The case $s=0$, which corresponds to empty monochromatic triangles, has been studied extensively over the last few years. In particular, it is known that $M_3(1, 0)=3$, $M_3(2, 0)=9$ and $M_3(c, 0)=\\infty$, for $c\\geq 3$. In this paper we extend these results when $c \\geq 2$ and $s \\geq 1$. 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