Tiling with monochromatic bipartite graphs of bounded maximum degree
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...
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| Main Authors: | , |
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| Format: | Article (Journal) |
| Language: | English |
| Published: |
[26 September 2024]
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| In: |
Mathematika
Year: 2024, Volume: 70, Issue: 4, Pages: e12280-1-e12280-22 |
| ISSN: | 2041-7942 |
| DOI: | 10.1112/mtk.12280 |
| Online Access: | Verlag, kostenfrei, Volltext: https://doi.org/10.1112/mtk.12280 Verlag, kostenfrei, Volltext: https://onlinelibrary.wiley.com/doi/abs/10.1112/mtk.12280 
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| Author Notes: | António Girão, Oliver Janzer |
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| 520 | |a 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. | ||
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