{"paper":{"title":"New lower bound on the Shannon capacity of C7 from circular graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.IT"],"primary_cat":"math.CO","authors_text":"Alexander Schrijver, Sven Polak","submitted_at":"2018-08-22T16:59:21Z","abstract_excerpt":"We give an independent set of size $367$ in the fifth strong product power of $C_7$, where $C_7$ is the cycle on $7$ vertices. This leads to an improved lower bound on the Shannon capacity of $C_7$: $\\Theta(C_7)\\geq 367^{1/5} > 3.2578$. The independent set is found by computer, using the fact that the set $\\{t \\cdot (1,7,7^2,7^3,7^4) \\,\\, | \\,\\, t \\in \\mathbb{Z}_{382}\\} \\subseteq \\mathbb{Z}_{382}^5$ is independent in the fifth strong product power of the circular graph $C_{108,382}$. Here the circular graph $C_{k,n}$ is the graph with vertex set $\\mathbb{Z}_{n}$, the cyclic group of order $n$,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.07438","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":""},"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"}