{"paper":{"title":"On \"stability\" in the Erd\\H{o}s-Ko-Rado theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jeff Kahn, Pat Devlin","submitted_at":"2015-02-19T20:32:42Z","abstract_excerpt":"Denote by $K_p(n,k)$ the random subgraph of the usual Kneser graph $K(n,k)$ in which edges appear independently, each with probability $p$. Answering a question of Bollob\\'as, Narayanan, and Raigorodskii,we show that there is a fixed $p<1$ such that a.s. (i.e., with probability tending to 1 as $k \\to \\infty$) the maximum independent sets of $K_p(2k+1, k)$ are precisely the sets $\\{A\\in V(K(2k+1,k)): x\\in A\\}$ ($x\\in [2k+1]$).\n  We also complete the determination of the order of magnitude of the \"threshold\" for the above property for general $k$ and $n\\geq 2k+2$. This is new for $k\\sim n/2$, wh"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.05692","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"}