{"paper":{"title":"The maximum number of complete subgraphs of fixed size in a graph with given maximum degree","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"A.J. Radcliffe, Jonathan Cutler","submitted_at":"2014-05-06T15:35:50Z","abstract_excerpt":"In this paper, we make progress on a question related to one of Galvin that has attracted substantial attention recently. The question is that of determining among all graphs $G$ with $n$ vertices and $\\Delta(G)\\leq r$, which has the most complete subgraphs of size $t$, for $t\\geq 3$. The conjectured extremal graph is $aK_{r+1}\\cup K_b$, where $n=a(r+1)+b$ with $0\\leq b\\leq r$. Gan, Loh, and Sudakov proved the conjecture when $a\\leq 1$, and also reduced the general conjecture to the case $t=3$. We prove the conjecture for $r\\leq 6$ and also establish a weaker form of the conjecture for all $r$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.1322","kind":"arxiv","version":1},"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"}