Ramsey properties for V-shaped posets in the Boolean lattice

In this paper, we study the Ramsey properties for V-shaped posets. A V-shaped posetVm,n can be obtained by identifying the minimal elements of two chains on m+1 and n+1 elements, respectively. Let P1,P2,…,Pk be posets. The Boolean Ramsey numberR(P1,P2,…,Pk), first introduced by Axenovich and Walzer [2], is the minimum number n such that no matter how we color the elements in the Boolean lattice Bn with k colors, there always exists a poset Pi contained in Bn whose elements are all colored with i. We investigate R(P1,P2,…,Pk) for given V-shaped posets as Pis and manage to determine this value in some cases. Next, we characterize the minimal posets Q contained in Bn, where n=R(P1,P2,…,Pk), having the Ramsey property described in the previous paragraph when Pi=V1,1 for all i or when k=2 and Pi=Vi,i for i=1,2. Given posets P and Q, we define the Boolean rainbow Ramsey numberRR(P,Q) as the minimum number n such that when arbitrarily coloring the elements in Bn, there always exists either a monochromatic P or a rainbow Q. An upper bound for RR(P,Ak) was given by Chang, Li, Gerbner, Methuku, Nagy, Patkos, and Vizer [4] for a general poset P and the k-element antichain Ak. We give the exact value of RR(Vm,n,Ak) if m≠n.