Geodesic
The Boolean rainbow Ramsey number of antichains, Boolean posets and chains
Hong-Bin Chen1 ; Yen-Jen Cheng2 ; Wei-Tian Li3 ; Chia-An Liu4
1National Chung Hsing University
2National Taiwan Normal University
3Department of Applied Mathematics, National Chung Hsing University
4Xiamen University Malaysia
The electronic journal of combinatorics, Tome 27 (2020) no. 4
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Motivated by the paper, Boolean lattices: Ramsey properties and embeddings Order, 34 (2) (2017), of Axenovich and Walzer, we study the Ramsey-type problems on the Boolean lattices. Given posets $P$ and $Q$, we look for the smallest Boolean lattice $\mathcal{B}_N$ such that any coloring of elements of $\mathcal{B}_N$ must contain a monochromatic $P$ or a rainbow $Q$ as an induced subposet. This number $N$ is called the Boolean rainbow Ramsey number of $P$ and $Q$ in the paper. Particularly, we determine the exact values of the Boolean rainbow Ramsey number for $P$ and $Q$ being the antichains, the Boolean posets, or the chains. From these results, we also derive some general upper and lower bounds of the Boolean rainbow Ramsey number for general $P$ and $Q$ in terms of the poset parameters.
DOI : 10.37236/9034
Classification : 05C55, 05D05, 05D10, 06A07, 06B75
Mots-clés : Boolean lattice, Ramsey-type problems

Hong-Bin Chen  1   ; Yen-Jen Cheng  2   ; Wei-Tian Li  3   ; Chia-An Liu  4

1 National Chung Hsing University
2 National Taiwan Normal University
3 Department of Applied Mathematics, National Chung Hsing University
4 Xiamen University Malaysia 
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     author = {Hong-Bin Chen and Yen-Jen Cheng and Wei-Tian Li and Chia-An Liu},
     title = {The {Boolean} rainbow {Ramsey} number of antichains, {Boolean} posets and chains},
     journal = {The electronic journal of combinatorics},
     year = {2020},
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     number = {4},
     doi = {10.37236/9034},
     zbl = {1453.05064},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/9034/}
}
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Hong-Bin Chen; Yen-Jen Cheng; Wei-Tian Li; Chia-An Liu. The Boolean rainbow Ramsey number of antichains, Boolean posets and chains. The electronic journal of combinatorics, Tome 27 (2020) no. 4. doi: 10.37236/9034

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