{"paper":{"title":"On almost Gallai colourings in complete graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alexandr Grebennikov, Let\\'icia Mattos, Tibor Szab\\'o","submitted_at":"2025-03-21T17:33:27Z","abstract_excerpt":"For $t \\in \\mathbb{N}$, we say that a colouring of $E(K_n)$ is $\\textit{almost}$ $t$-$\\textit{Gallai}$ if no two rainbow $t$-cliques share an edge. Motivated by a lemma of Berkowitz on bounding the modulus of the characteristic function of clique counts in random graphs, we study the maximum number $\\tau_t(n)$ of rainbow $t$-cliques in an almost $t$-Gallai colouring of $E(K_n)$. For every $t \\ge 4$, we show that $n^{2-o(1)} \\leq \\tau_t(n) = o(n^2)$. For $t=3$, surprisingly, the behaviour is substantially different. Our main result establishes that $$\\left ( \\frac{1}{2}-o(1) \\right ) n\\log n \\l"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2503.17334","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2503.17334/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"}