{"paper":{"title":"On the directions determined by Cartesian products and the clique number of generalized Paley graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Chi Hoi Yip","submitted_at":"2020-10-05T05:12:10Z","abstract_excerpt":"It is known that the number of directions formed by a Cartesian product $A \\times B \\subset AG(2,p)$ is at least $|A||B| - \\min\\{|A|,|B|\\} + 2$, provided $p$ is prime and $|A||B|<p$. This implies the best known upper bound on the clique number of the Paley graph over $\\mathbb{F}_p$. In this paper, we extend this result to $AG(2,q)$, where $q$ is a prime power. We also give improved upper bounds on the clique number of generalized Paley graphs over $\\mathbb{F}_q$. In particular, for a cubic Paley graph, we improve the trivial upper bound $\\sqrt{q}$ to $0.769\\sqrt{q}+1$. In general, as an applic"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2010.01784","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2010.01784/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}