{"paper":{"title":"Constructive Winning Breaker Strategies in the Maker-Breaker $C_k$-Game","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Anand Srivastav, Matthias Sowa","submitted_at":"2026-07-01T13:22:03Z","abstract_excerpt":"Maker-Breaker subgraph games are among the most famous combinatorial games. For $n,q\\in\\mathbb{N}$ and a fixed subgraph $C$ of the complete graph $K_n$, the two players, called Maker and Breaker, alternately claim edges of $K_n$. Maker claims one unclaimed edge per round and Breaker may claim up to $q$ edges per round. If Maker is able to claim all edges of a copy of $C$, he wins the game. Otherwise Breaker wins. Bednarska and {\\L}uczak (2000) determined in a landmark work the asymptotics of the treshold bias as $\\Theta(n^{1/m(C)})$ where $m(C)$ is the 2-density of $C$, analysing random strate"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.01294","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.01294/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}