Tiling with monochromatic bipartite graphs of bounded maximum degree

• Verfasst von: Girao, Antonio [VerfasserIn]
• Verfasst von: Janzer, Oliver [VerfasserIn]
• Titel: Tiling with monochromatic bipartite graphs of bounded maximum degree
• Verf.angabe: António Girão, Oliver Janzer
• E-Jahr: 2024
• Jahr: [26 September 2024]
• Umfang: 22 S.
• Fussnoten: Gesehen am 27.02.2025
• Titel Quelle: Enthalten in: Mathematika
• Ort Quelle: Hoboken, NJ : Wiley, 1954
• Jahr Quelle: 2024
• Band/Heft Quelle: 70(2024), 4, Artikel-ID e12280, Seite e12280-1-e12280-22
• ISSN Quelle: 2041-7942
• DOI: doi:10.1112/mtk.12280
• Datenträger: Online-Ressource
• Sprache: eng
• K10plus-PPN: 1918743339

Abstract

We prove that for any r∈N\rın \mathbb N\, there exists a constant Cr\C_r\ such that the following is true. Let F=F1,F2,⋯\mathcal F=łbrace F_1,F_2,\dots \rbrace\ be an infinite sequence of bipartite graphs such that |V(Fi)|=i\|V(F_i)|=i\ and Δ(Fi)⩽Δ\Delta (F_i)łeqslant \Delta\ hold for all i\i\. Then, in any r\r\-edge-coloured complete graph Kn\K_n\, there is a collection of at most exp(CrΔ)\exp (C_r\Delta)\ monochromatic subgraphs, each of which is isomorphic to an element of F\mathcal F\, whose vertex sets partition V(Kn)\V(K_n)\. This proves a conjecture of Corsten and Mendonça in a strong form and generalises results on the multi-colour Ramsey numbers of bounded-degree bipartite graphs. It also settles the bipartite case of a general conjecture of Grinshpun and Sárközy.