{"paper":{"title":"The approximate Loebl-Koml\\'os-S\\'os Conjecture I: The sparse decomposition","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Diana Piguet, Endre Szemer\\'edi, Jan Hladk\\'y, J\\'anos Koml\\'os, Maya J. Stein, Mikl\\'os Simonovits","submitted_at":"2014-08-17T21:02:52Z","abstract_excerpt":"In a series of four papers we prove the following relaxation of the Loebl-Komlos-Sos Conjecture: For every $\\alpha>0$ there exists a number $k_0$ such that for every $k>k_0$ every $n$-vertex graph $G$ with at least $(\\frac12+\\alpha)n$ vertices of degree at least $(1+\\alpha)k$ contains each tree $T$ of order $k$ as a subgraph.\n  The method to prove our result follows a strategy similar to approaches that employ the Szemer\\'edi regularity lemma: we decompose the graph $G$, find a suitable combinatorial structure inside the decomposition, and then embed the tree $T$ into $G$ using this structure."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.3858","kind":"arxiv","version":3},"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"}