For each integer $n,m \ge 1$

$2n \le R(Q_n,Q_n) \le n^2+2n$.

Here $Q_n$ denotes a Boolean lattice of dimension $n$ and $R(P,P')$ or $R_{\dim_2}(P,P')$ is the smallest $N$ such that any red/blue coloring of $Q_N$ contains either red copy or $P$ or a blue copy of $Q$. (It is called a poset Ramsey number.)

For the lower bound on $R(Q_n,Q_n) $, with $Q_n$ consider $Q_{2n-1}$. Color the sets of size $0, . . .,n-1$ red and all other sets blue. Then there is no monochromatic chain with $n+1$ elements and there is no monochromatic copy of $Q_n$. it's easy to understand the lower bound and I am stuck to understand the upper bound.

How do you find an upper bound for $n^2+2n$? Any help will be appreciated. Thanks.

I found this result in the paper Maria Axenovich, Stefan Walzer: *Boolean lattices: Ramsey properties and embeddings*, https://arxiv.org/abs/1512.05565, http://dx.doi.org/10.1007/s11083-016-9399-7

The proof given in the paper is:

For the upper bound on $R(Q_n,Q_n)$, consider a red/blue coloring of $Q_{n^2+2n}$. Let the ground set be $X_0\cup X_1 \cup \dots \cup X_{n+1}$, where $X_i$'s s are pairwise disjoint and of size $n$ each. Consider families of sets $\mathcal B_Y$ for each $Y\subseteq X_0$ with $|Y|\ge1$ to be $\mathcal B_Y = \{Y\cup X_1\cup \dots \cup X_{|Y|} \cup X; X\subseteq X_{|Y|+1}\}$, let $\mathcal B_{\emptyset}=2^{X_1}$. We see that each $\mathcal B_Y$ is a copy of $Q_n$. If this copy is blue, then $\mathcal B_Y$ gives a monochromatic copy of $Q_n$. Otherwise, there is a red element in each $\mathcal B_Y$. This element is $Z_Y=Y\cup X_1 \cup \dots X_{|Y|}\cup S_Y$, where $S_Y\subseteq X_{|Y|+1}$. We claim that these elements form a red copy of $Q_n$. Indeed, we see for $Y, Y' \subseteq [n]$ that $Y\subseteq Y'$ iff $Z_Y\subseteq Z_{Y'}$.