{"paper":{"title":"On the average size of independent sets in triangle-free graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Barnaby Roberts, Ewan Davies, Matthew Jenssen, Will Perkins","submitted_at":"2016-06-03T10:59:07Z","abstract_excerpt":"We prove an asymptotically tight lower bound on the average size of independent sets in a triangle-free graph on $n$ vertices with maximum degree $d$, showing that an independent set drawn uniformly at random from such a graph has expected size at least $(1+o_d(1)) \\frac{\\log d}{d}n$. This gives an alternative proof of Shearer's upper bound on the Ramsey number $R(3,k)$. We then prove that the total number of independent sets in a triangle-free graph with maximum degree $d$ is at least $\\exp \\left[\\left(\\frac{1}{2}+o_d(1) \\right) \\frac{\\log^2 d}{d}n \\right]$. The constant $1/2$ in the exponent"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.01043","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"}