Erd\H{o}s-Gy\'{a}rf\'{a}s problem for partially ordered sets

Given integers $p,q,t$ with $1 \le t \le p$ and $1 \le q \le h_p(t)$, a strong $(p,q,t)$-coloring of the Boolean lattice $B_n$ is a coloring of its $t$-chains such that every induced copy of $B_p$ in $B_n$ uses at least $q$ colors on its $t$-chains. Let $f_t^{\sharp}(n,p,q)$ denote the minimum number of colors in such a coloring. We study this Boolean-lattice analogue of the Erd\H{o}s-Gy\'{a}rf\'{a}s function.We first show that every finite poset strongly embeds into a Boolean lattice. Combined with a structural Ramsey theorem for finite posets with linear extensions, this implies the existence of the strong Boolean Ramsey number $\mathrm{R}^{\sharp}_{k,t}(\mathcal{B}\mid Q)$ for every integer $k\ge1$, every $t\ge1$, and every nonempty finite poset $Q$. In particular, this gives an affirmative answer to a problem of Cox and Stolee and yields the existence of $f_t^{\sharp}(n,p,2)$. Next, using the symmetric Lov\'asz local lemma, we obtain a probabilistic upper bound on $f_t^{\sharp}(n,p,q)$. Finally, we prove lower bounds by combining Tur\'an-type extremal estimates for $t$-chains, a double-counting argument, and a generalized Lubell-type framework for $t$-chains.