The Boolean Rainbow Ramsey Number of Antichains, Boolean Posets and Chains

Hong-Bin Chen, Yen-Jen Cheng, Wei-Tian Li, Chia-An Liu
2020 Electronic Journal of Combinatorics  
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
more » ... 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 fatcat:d2x6l5jugfgs7jr2rgtt2rfag4