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

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 B-N such that any coloring of elements of 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.