{"paper":{"title":"On Small Folkman Graphs Arrowing $K_2$ or $K_3$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Stanis{\\l}aw Radziszowski, Steven Van Overberghe, Zohair Raza Hassan","submitted_at":"2026-05-15T18:37:43Z","abstract_excerpt":"For a graph $G$ and integers $a_i \\geq 1$, we say that $G \\xrightarrow[]{} (a_1, \\ldots, a_k)^v$ if in any $k$-coloring of $G$'s vertices there exists a monochromatic $a_i$-clique for some color $i \\in \\{1,\\ldots,k\\}$. $G \\xrightarrow[]{} (a_1, \\ldots, a_k)^e$ is defined similarly, but for edge colorings. The Folkman number $F_v(a_1, \\ldots, a_k; H)$ is the smallest number of vertices for which an $H$-free graph arrowing $(a_1, \\ldots, a_k)^v$ exists. $F_e(a_1, \\ldots, a_k; H)$ is defined similarly for edge-arrowing.\n  In this work, we present new bounds for Folkman numbers where $a_i \\in \\{2,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.16542","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/2605.16542/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"claim_evidence","ran_at":"2026-05-19T19:21:56.914550Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T18:33:26.639584Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"301be1fff96d9c7b80f5407e84fb997bc70fc2cd43e190e478da7a35f4d5224e"},"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"}